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# The multi-faceted reach of the OEIS

### From OeisWiki

The OEIS covers more than just math: it has entries that go into various scientific fields and even the arts. To quote Brian Hayes on the OEIS:^{[1]}

- Number theory and combinatorics are naturally well represented, but there are also lots of examples from switching theory and circuit design (combinations of Boolean functions), chemistry (numbers of alkanes, ethers, esters, etc., with n carbon atoms) and a few from physics (Feynman diagrams with n vertices) and biology (secondary structures of RNA with n nucleotides).

This is an attempt to list some of those fields and the relevant material.

## Natural sciences

### Life sciences

#### Biology

A number of combinatorial structures (A001147, A000111, A001190, A006472, A094503) have applications in population genetics. Some sequences can be interpreted as modeling population dynamics, like A196448. A008792, the amino acid numbers, is a mathematical representation of amino acids in theoretical biology.

##### Botany

###### Phyllotaxy

In botany, dynamical systems or their discretizations are often used in growth modeling, especially phyllotaxis (plant morphology), following the work of Irving Adler; see A093613, A002965, A036299, A122771, A122773, A138201, and A139344.

The Fibonacci numbers (A000045) appear in the phyllotaxis of sunflowers and pineapples, and also in the leaf arrangements of many plants.

Even combinatorial game theory is relevant to phyllotaxis, see A047708.

##### Zoology

##### Genetics

- A000311 Number of phylogenetic trees with n nodes.
- A074206 Lower bound on the worst-case number of solutions to the probed partial digest problem for n fragments of DNA.
- A088518 Symmetric secondary structures of RNA molecules with n nucleotides.
- A104155 The 64 codons of the genetic code, giving the value 1 to thymine (T), 3 to adenine (A), 2 to cytosine (C) and 4 to guanine (G).
- A164057 Complement to A164056, change A164056 bits (0->1; 1->0). Provides a coding template for Petoukhov matrices, relating to DNA codons.
- A180850 Number of codons that code for an amino acid, listed in alphabetical order of their single-letter codes.
- A202478 Decimal expansion of constant arising in combinatorics of gamma-structures (RNA pseudoknot related).
- A000918 Time since a common genetic ancestor.
^{[2]}

##### Microbiology

A155940 relates to the "condom problem" (not a practical strategy!).

###### Virology

A003136 and A034017 are related to the number of structural sub-units of icosahedral viruse capsids.

- A101451 Location of the restriction sites for the enzyme BccI on the PhiX174 DNA.
- A108749 Location of the restriction sites for the enzyme BceA1I in PhiX174 DNA.
- A108776 Location of the restriction sites for the enzyme BseMII in PhiX174 DNA.
- A108785 Location of the restriction sites for the enzyme BsuRI in PhiX174 DNA.
- A108875 Location of restriction sites for the enzyme LweI in PhiX174 DNA.
- A108876 Location of restriction sites for the enzyme MboII in PhiX174 DNA.
- A108877 Location of restriction sites for the enzyme PfeI in PhiX174 DNA.
- A108878 Location of restriction sites for the enzyme SchI in PhiX174 DNA.

##### Proteomics

- A000217 Number of ways a peptide of n amino acid residues can be broken up in a mass spectrometer.

### Physical sciences

#### Chemistry

A number of sequences are related to chemical constants, e.g.

- A070062 Decimal expansion of old estimate for Avogadro's constant.
- A087778 Decimal expansion of Avogadro's constant.
- A129106 Decimal expansion of proposed value for Avogadro's number, namely 602214141070409084099072 = 84446888^3.

A large number of sequences enumerate molecules of various types, e.g.

- A000598 Alkyl radicals C_n H_{2n+1} ignoring stereoisomers.
- A000600 Tertiary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH).
- A000602 n-carbon alkanes C_n H_{2n+2)} ignoring stereoisomers. (For our Dutch speakers, the blog article Huiswerk (met bijna twintig jaar vertraging) relates a story about finding this sequence in chemistry class.)
- A130753 Related to CnH2*n+2 straight chain alkanes.
- A018190 Number of benzenoid hydrocarbons with n hexagons.
- A067608 Structural alkanes with combinatorial diameter n.
- A130016 Number of isomers for different number of substitutions of hydrogen atoms by deuterium in the chair form of cyclohexane.
- A219863 Fraction of unjoined substituents on an infinite substrate (Flory's example is polyvinyl alcohol).

##### Thermochemistry

##### Electrochemistry

##### Inorganic chemistry

Perhaps the most fundamental sequence in chemistry is

- A080915 Number of electrons in outermost electron shell (valence electrons) in chemical element number n.

Many sequences relate to the periodic table (Mendeleev's periodic table or Janet's periodic table) of the chemical elements, e.g.

- A093907 Number of elements in the n-th period of the periodic table.
- A018227 Magic numbers: atoms with full [valence] shells containing any of these numbers of electrons are considered electronically stable.
- A167268 Janet's sequence. Number of elements for each successively filled electronic subshells of an atom.
- A101358 Electron affinity numbers (atomic numbers corresponding to the electron affinity).
- A138040 Atomic number of the 2nd transition metal (group 4) in the n-th row of the periodic system of elements.
- A173642 A distribution of electrons. By rows whose sum is A000027 (for periodic table).
- A180638 Phan Thành Nam's upper bound on the number of non-relativistic electrons bound to a nucleus of charge n.

A number of sequences are related to the properties of chemical elements, e.g.

- A007656 Atomic weights of the elements.

A number of sequences are related to atomic structure, e.g.

- A080916 Number of electrons in outermost electron subshell in chemical element number n.
- A033177 Number of distinct distances between n electrons in minimal energy configuration on a sphere.
- A168208 Table of the number of electrons of the n-th element of the PSE in atomic shells.
- A249947 Number of available orbitals at increasing subshells in multi-electron atoms.

A number of sequences are related to molecular structure, e.g.

- A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule.

##### Organic chemistry

- A005960 Number of paraffins with n carbon atoms.

##### Analytical chemistry

###### Stoichiometry

###### Chromatography

###### Spectroscopy

- A081821 Rydberg constant.
- A126252 Wavenumbers of red, turquoise, blue, indigo and violet in the spectrum of hydrogen, as first measured by Robert Bunsen and Gustav Kirchhoff in 1859.

###### Crystallography

#### Physics

##### Classical physics

###### Classical mechanics

In fluid dynamics, the constant A197688 is the pressure drag coefficient for Kirchhoff flow past an infinite plate.

###### Classical gravity

- A070058 Decimal expansion of Newton's gravitational constant.
- A125125 Decimal expansion of the Earth's gravitational constant (mass of Earth's atmosphere included) of the World Geodetic System 1984 Ellipsoid, second upgrade.
- A208745 Decimal expansion of the gravitoid constant.
- A230242 Mass ratio for stable L4 and L5 Lagrange points.

###### Statistical mechanics

- A213054 Decimal expansion of first Chandrasekhar's nearest neighbor constant.
- A213055 Decimal expansion of second Chandrasekhar's nearest neighbor constant.
- A001393 High temperature series for spin-1/2 Ising free energy on 3-dimensional simple cubic lattice.

###### Thermodynamics

- A070063 Decimal expansion of Boltzmann's constant.
- A081820 Stefan-Boltzmann constant.
- A070064 Decimal expansion of the molar gas constant.
- A137778 Triangular sequence from coefficients of an expansion of a Rankine-Hugoniot relation function for density in terms of thermodynamic gamma as

and pressure ratio as*t*

:*x**p*(*x*,*t*) =

.( *t*+ 1) / (*t*− 1) +*x*1 + *x*(*t*+ 1) / (*t*− 1)

- A256460 Triple point of water.

###### Classical electrodynamics

- A081799 Decimal expansion of electric permittivity constant epsilon_0.
- A019694 Decimal expansion of magnetic permeability constant mu_0. (Decimal expansion of 2*Pi/5.)
- A163999 Decimal expansion of the Faraday constant.

###### Optics

- A245461 Constant in the Rayleigh criterion: first zero of J_1, divided by Pi.

###### Special relativity

- A003678 Decimal expansion of speed of light (meters/second).
- A182999 Decimal expansion of the integer c^2 where c = 299792458 (exactly) is the speed of light in vacuum (m/s).

###### General relativity

##### Quantum physics

- A003676 Decimal expansion of Planck constant (joule sec).
- A201488 Maximal success probability of the CHSH game.
- A051862 Perturbation expansion in quantum field theory: scalar case in 6 spacetime dimensions.
- A261400 Quantum knots
- A171545 The absolute value of the Clebsch-Gordan coupling coefficient <j1 j2; m1 m2 | J M> = <2 3/2 ; 0 -3/2 | 7/2 -3/2>.

A number of sequences are related to Planck units, e.g.

- A003676 Decimal expansion of Planck constant (joule sec).
- A078300 Decimal expansion of Planck length.
- A078301 Decimal expansion of Planck mass.
- A078302 Decimal expansion of Planck time.
- A131223 Bekenstein bound in natural (Planck) units.
- A210491 Decimal expansion of Planck temperature.
- A228817 Decimal expansion of Planck force: F_P = c^4/G, in SI units.
- A235995 Decimal expansion of Planck angular frequency in Hertz with six proved digits: 1.85492×10^43 Hz.
- A242978 Decimal expansion of Efimov's scaling constant.
- A246504 Decimal expansion of Planck charge in coulombs.
- A246505 Decimal expansion of Planck area in square meters.
- A254181 Decimal expansion of reduced Planck constant in J s.
- A255896 Decimal expansion of Planck energy in Joules.

###### Quantum electrodynamics (QED)

- A195022 Negative of the decimal expansion of the electron spin g-factor.

A number of sequences are related to the fine structure constant and its reciprocal, e.g. A003673, A005600 and A082726.

A number of sequences are related to Feynman diagrams, e.g.

- A115974 Number of Feynman diagrams of the proper self-energy at perturbative order n (...).
- Feynman diagrams of order 2n. (Cf. A005411, A005411, A005412, A005413, A005414)
- A122023 Vacuum Virtual Particle 10 vertex graph as Feynman diagram seen as (...).
- A167872 A sequence of moments connected with Feynman numbers (A000698): Half the number of Feynman diagrams of order 2(n+1), for the electron self-energy in quantum electrodynamics (QED), i.e. all proper diagrams including Furry vanishing diagrams (those that vanish in 4-dimensional QED because of Furry theorem).

###### Electroweak theory

- A002978 'Susceptibility series for honeycomb.'
*- Possibly belongs here.*

###### Quantum chromodynamics (QCD)

- A181635 Unmatter combinations of quarks and antiquarks of length n>=1 that preserve the colorless.
- A181633 Unmatter combinations as pairs of quarks (q) and antiquarks (a), (...)
- A181634 Colorless combinations as pairs of quarks and antiquarks (...)
- A181685 Colorless combinations of quarks and antiquarks of length n >= 1.

A number of sequences discuss quark and gluon diagrams.

###### Standard Model

###### Quantum gravity

- A245494 Constant in the third-order correction term for gravitational potential in natural units, negated.

###### Representation theory

Isotropic quantum harmonic oscillators have an infinite number of bound states, which map surjectively to the natural numbers. It is possible to enumerate many interesting degeneracy sequences, especially when considering anharmonic perturbations. After Bethe, we find that: Many Correlation Tables are Molien Sequences. Detailed calculations available at Wolfram Demonstrations explain connections to the following sequences: A000004, A000012, A000027, A000035, A000217, A000292, A000332, A000389, A002264, A002265, A002266, A004396, A004523, A005044, A008613, A008615, A008616, A008642, A008647, A008651, A008672, A008676, A008679, A008720, A008731, A008733, A008743, A051274, A051275, A059169, A091972, A112689, A181286, A254011.

##### String theory

- A023020 (conjecturally) appears in the series expansion of the partition function in the system on the cone pure spinors.

##### Particle physics

- A081803 Decimal expansion of Compton electron radius : r_c = 3.861593228000001.10-13 (m).
- A081802 Decimal expansion of electron charge 1.602177330000*10-19 (Coulomb).
- A081823 Electron volt eV in J (also elementary charge in C ).
- A081813 Atomic mass unit-electron volt relationship : 931.494013 * 10^6 eV.
- A063516 Multiples of single electron masses yielding stable particles intermediate between an electron and a proton as given in reference below.
- A107270 Multiples of coefficients in asymptotic expansion of the rotational partition function for a heteronuclear diatomic molecule.

A number of sequences are related to the properties of elementary particles, e.g.

- A225359 Decimal expansion of suggested value for N_eff: the effective number of neutrino species present in the era before recombination.

- A070060 Mass of an electron (in kg). (A070060 is
**NOT QUITE DUPLICATE (VALUE DIFFERS) of A081801**) - A081801 Decimal expansion of electron mass in kg. (A081801 is
**NOT QUITE DUPLICATE (VALUE DIFFERS) of A070060**) - A003672 Decimal expansion of electron mass (mass units).

- A?????? Mass of a muon (in kg).
- A254291 Decimal expansion of muon mass (in kg).
- A057720 Decimal expansion of muon-to-electron mass ratio.

- A?????? Mass of a tauon (in kg).

- A070059 Mass of a proton (in kg).
- A003677 Decimal expansion of proton mass (mass units).
- A005601 Decimal expansion of proton-to-electron mass ratio.

- A?????? Mass of a neutron (in kg).
- A003675 Decimal expansion of neutron mass (mass units).
- A006833 Decimal expansion of neutron-to-electron mass ratio.
- A006834 Decimal expansion of neutron-to-proton mass ratio.

##### Nuclear physics

A number of sequences are related to the structure of atomic nuclei, e.g.

- A130598 A shell geometric model of the nucleus. The location of the magic numbers. A triangle.
- A018226 Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay.
- A046939 "Magic numbers" for the number of neutrons in nucleus.

A number of sequences are related to radioactive decay, e.g.

- A080533 Number of protons in longest known radioactive decay series ending with Lead 206 ("uranium series"), reversed.
- A080534 Number of protons in longest known radioactive decay series ending with Lead 207 ("actinium series"), reversed.
- A080535 Number of protons in longest known radioactive decay series ending with Lead 208 ("thorium series"), reversed.
- A080536 Number of protons in longest known radioactive decay series ending with Lead 209, reversed.
- A080537 Number of neutrons in longest known radioactive decay series ending with Lead 206 ("uranium series"), reversed.
- A080538 Number of neutrons in longest known radioactive decay series ending with Lead 207 ("actinium series"), reversed.
- A080539 Number of neutrons in longest known radioactive decay series ending with Lead 208 ("thorium series"), reversed.
- A080540 Number of neutrons in longest known radioactive decay series ending with Lead 209, reversed.

##### Astrophysics

- A268682 Maximum percentage of mass-energy of a black hole which can come from angular momentum, and hence the maximum energy which can be extracted from the black hole via the Penrose process.
- A268683 Maximum increase in mass-energy a particle can carry away from a neutral rotating (Kerr) black hole via the Penrose process.

##### Cosmology

- A122505 Arises from energy spectrum of three dimensional gravity with negative cosmological constant, in analysis by Edward Witten.

### Earth and space sciences

#### Astronomy

From Earth, Mercury and Venus can sometimes be seen going across the Sun. Those are somewhat rare events (more so for Venus than Mercury). A171466 and A171467 list years in which transits of Mercury and Venus respectively have taken place or are expected to occur.

The number of stars of particular brightness is the subject of A053406 and A072171, though this is perhaps not well-defined.

A072235 and A100748 give years of recorded appearances of Halley's comet.

- A003461 Bode numbers multiplied by 10.
- A070273 Number of letters in the English names of the planets (in order from smallest orbit to largest).
- A116448 Rotational periods (length of year) of planets in the solar system, to the nearest whole number of terrestrial days.
- A118652 Diameters in miles of the planets in the solar system, starting with the closest to the sun.
- A122559 Position of first letter of n-th planet (in English) in alphabet.
- A130788 Solar planets' average orbit velocity (inverse ratio relative to one of Mercury), multiplied by 3 and rounded to the nearest integer.
- A131500 Radii of planets in solar system, in units of radius of Mercury, multiplied by 4.
- A189824 Decimal expansion of Pogson's ratio 100^(1/5).
- A209257 A musically inspired Titius-Bode-like sequence.
- A212002 Constant 4*Pi^2 from Kepler's 3rd Law, .
- A227952 Wow! signal intensity signature (6EQUJ5).
- A248747 The Arecibo message written as binary string: a "0" is represented by an "off" radio pulse, while a "1" is represented by an "on" radio pulse.

#### Space exploration

- A247849 Apollo missions that landed humans on the moon.

#### Atmospheric and oceanic sciences

- A097403 Minimum wind speed in knots for Beaufort number n.
- A098516 Minimum wind speed in mph for a category n hurricane in the Saffir-Simpson hurricane scale.
- A097404 Minimum wind speed in statute miles per hour for Beaufort Number n.
- A098517 Minimum wind speed in mph for a tornado of Fujita (or Fujita-Pearson) scale F_n.

#### Geology

#### Physical geography

Number of planar maps without loops or isthmuses: A006398, A006399.

- A064296 Heights of 8000 meter peaks in meters (as of Sep 25 2001).

## Formal sciences

### Computer science

### Logic

Many OEIS sequences relate to formal logic. For example, how complex can an n-input Boolean function be? A056287 and A178939 give the cases of AND/OR and AND/OR/XOR complexities.

- A140861 Decimal Gödelization of Heyting's 11 axioms for intuitionistic propositional logic.
- A000231 Number of inequivalent Boolean functions of n variables under action of complementing group.

### Mathematics

There is no need to reinforce the reach of the OEIS into number theory and combinatorics. But the OEIS does touch into other branches of mathematics, even those branches where experts proclaim "the death of numbers."

#### Algebra

#### Calculus

#### Category theory

#### Game theory

#### Geometry

#### Knot theory

- A002863 Number of prime knots with n crossings.
- A002864 Number of alternating prime knots with n crossings.
- A018240 Rational knots with n crossings (up to mirroring).
- A052415 Number of n-crossing knots having symmetry group D1.
- A051450 Number of positive rational knots with 2n+1 crossings.
- A078477 Number of knots with n crossings and rational unknotting number = 1 (chiral pairs counted only once).
- A091518 Decimal expansion of the hyperbolic volume of the figure eight knot complement.
- A067640 Table T(n,k) giving number of two-legged knot diagrams with n >= 0 self-intersections and k >= 0 tangencies, read by antidiagonals.
- A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).
- A058031 Values of n^4-2*n^3+3*n^2-2*n+1, the Alexander polynomial for common knots.
- A172293 Number of prime knots up to nine crossings with determinant 2n+1 and signature 0.

#### Probability

#### Set theory

#### Topology

#### Recreational mathematics

### Statistics

Poisson distribution: Stirling numbers of the second kind (A008277) are used to compute the moments of a Poisson-distributed variable (see also A001861); the Bell numbers A000110 give the special case . A000262 can be used to compute the ascending factorial moments. For the median of a Poisson-distributed variable, see A259284. Some sequences which are modeled as Poisson-distributed include A000110, A001223, A079267, A185025, A185070. Other related sequences include A103647, A202357, A100124, A183299, A264806.

Exponential distribution: The Lah numbers (A008297) are used to compute the moments of the sum of independent exponentially-distributed variables. Computing the second moments requires the subfactorial numbers A000166. See also A246823.

Geometric distribution: The raw moments of the geometric distribution with are given by A000629/2.

Wilcoxon signed-rank test: A226329, A226330, A226331, A226332, A008967, A008968, A053632; see also A000601, A002620, A002621, A002622, A002623, A002624, A002625, A002626, A259324, A259325 relating to Fix & Hodges 1955.

A large number of sequences appeal to concepts in probabilistic number theory such as the Cramér random prime model (A235402, A235492), the Gauss–Kuzmin distribution (A084587, A241773, A242053, A084576, A084577, A084579-A084586; A038517 is the Gauss-Kuzmin-Wirsing constant), or the Erdős–Kac theorem (A070962).

### Systems science

#### Complex systems

#### Cybernetics

#### Dynamical systems

#### Operations research

- A155940 Triangle read by rows containing Vardi's optimal solution to the glove problem.

## Applied sciences and technology

### Engineering

#### Aerospace engineering

The constant A197688 is the pressure drag coefficient for Kirchhoff flow past an infinite plate or wing, used in aeroacoustics and computational fluid dynamics in general.

#### Civil engineering

#### Computer engineering

There are a great many sequences dealing with logic circuits (e.g., A002631, A002632, A059687).

#### Electrical engineering

A number of sequences are related to electrical networks, e.g.

- A002631 Number of circuits of nullity n.
- A002632 Number of circuits of rank n.
- A001677 Number of series-parallel networks with n edges.
- A211074 Equivalent resistance between two nodes on an infinite rectangular lattice of ideal unit resistors, where the nodes are separated by two resistors along one axis and one resistor on the other.

#### Electronic engineering

A number of sequences are related to digital displays of numbers with 7 segments, e.g.

- A010371 Number of segments needed to represent n on calculator display (second version).
- A165244 The numbers commonly displayed with 7 segments in electric clocks, in ascending order of number of segments lit.

A number of sequences are related to electronic components, e.g.

- A072198 E12 range of preferred resistor values in electronic engineering.
- A072554 E24 range of preferred resistor values in electronic engineering.

#### Industrial design

- A110186 Strongly rounded Renard series R"20 of preferred numbers based on 10^(1/20).

#### Mechanical engineering

- A184938 Bolt tightening order for mechanical rings
- A214395 Decimal expansion of Betz's constant: maximum efficiency of a turbine.

#### Naval engineering

- A097403 Minimum wind speed in knots for Beaufort Number n.
- A214395 Decimal expansion of Betz's constant: maximum efficiency of a screw propeller.

#### Nuclear engineering

### Health sciences

#### Dietetics

- A094062 . (Related to the Camel-Banana Problem.)

#### Epidemiology

#### Medicine

#### Pharmacy

- A099600 Inventory application: Numerator of the probability (...) of at least one shortage per month when n customers each request one item (...)

#### Dentistry

- A119247 Contains the ISO human tooth numbering sequence

## Social sciences

### Anthropology

The number of ways to classify objects, A005646, was studied in anthropology to give a baseline against which to compare to determine if supposed patterns found were necessary or contingent.

### Demography

- A131271 Ranks in natural order of 2^n increasing real numbers appearing in limit cycles of interval iterations, or Median Spiral Order.

### Economics

Military engineer Vauban gave sequence A224749 as an example of the exponential growth of food supply, specifically pigs.

### History

- A192994 Numbers above 100 in the Fundamentalis Tabula Arithmeticae
- A209186 Admission order to the USA of the n-th state in alphabetical order.
- A219022 Āryabhaṭa's sine table. (Also Aryabhata or आर्यभट.)
- A253810 The life of Diophantus.

See Historical sequences.

#### Archeology

A240136 and A267693 relate to the Antikythera mechanism discovered in a shipwreck off the Greek island of Antikythera.

An icosahedron-shaped Mughal gold box from the treasury of Tipu Sultan, dating back a three or four centuries, was inscribed with twenty numbers (A124250, see also A124251). A much older Ishango bone was discovered with sixteen numbers marked on it (A100000; see also A200066).

The Rhind Mathematical Papyrus involves many calculations using sums of reciprocal integers, see A094871.

- A153646 Sum of the letter values of the words of the Sator square: SATOR AREPO TENET OPERA ROTAS.

### Linguistics

- A007477 Number of plausible parsings of "Buffalo
^{n}" sentences in English. - A213705 Number of plausible parsings of the sentence "Etsivät
^{n+1}" in Finnish.

### Political science

- A153634 Number of letters in the n-th word of the Preamble to the U.S. Constitution.
- A090232 First year in office of n-th President of the United States of America.

### Political geography

- A091786 Number of electoral votes per state in the U.S.A. (for 2001-2010).
- A112506 Number of counties per state of the USA in alphabetical order by state.
- A112507 Number of counties per state of the USA in increasing order.

The OEIS contains several sequences generalizing the Four color theorem:

- A000934 Chromatic number (or Heawood number) Chi(n) of surface of genus n.
- A000703 Chromatic number (or Heawood number) of nonorientable surface with n crosscaps.
- A230628 Maximum number of colors needed to color a planar map of several empires, each empire consisting of n countries.

### Psychology

One of the symptoms of Alzheimer's disease is trouble with short term memory. To test the short term concentration of patients, psychologists sometimes use the "serial sevens" test, in which the patient is asked to count backwards from 100 in steps of 7 (A115020).

Some psychologists study the difficulty of learning particular Boolean concepts; see the reference in A039754.

### Sociology

- A058992 Gossip Problem: there are n people and each of them knows some item of gossip not known to the others. They communicate by telephone and (...)

## Business and finance

- A109065 Numerator of the fraction due in month n of the total interest for a one-year installment loan based on the Rule of 78s (Each denominator is 78).
- A114139 Changes in United States postal rates per ounce since 1863.
- A182086 Number of ways of making change for n Pfennig using Deutschmark coins.
- A193616 Integers often used as card values in Planning Poker decks.
- A193622 Integers originally used as card values in Planning Poker.
- A208953 Amounts (in cents) of coins in denominations suggested by Shallit.
- A212950 Amounts (in cents) of Canadian coins in denominations suggested by Shallit.

See also sequences offering monetary rewards in the Index.

## Recreation and games

### Games

#### Tabletop games

##### Board games

A number of sequences are related to the game of Risk, e.g.

- A167179 The number of additional armies one receives in Parker Brothers' (now part of Hasbro) game of Risk for turning in the n-th set of three different or alike cards.

Chess is extremely well-represented in the OEIS, with hundreds of sequences. Perhaps the most famous is A000170, the *n*-queens problem. See 'chess' in the Index.

Japanese chess, also known as shōgi, also offers inspiration for new sequences, see A062103. Many of the sequences that have been computed for Chess, could be computed for shogi as well.

The game of Go is represented for example by A007565, A048289, A089071, A094777, A096259, and A261491.

Scrabble: A080993, A080994, A113172, A124015.

A number of sequences are related to the Solitaire Army game, e.g.

- A014225 Number of initial pieces needed to reach level n in the Solitaire Army game.
- A014227 Minimal number of initial pieces needed to reach level n in the Solitaire Army game on a hexagonal lattice (a finite sequence).
- A125730 Minimal number of initial pieces needed to reach level n in the Solitaire Army game when diagonal jumps are allowed.

##### Card games

Sequences related to Mousetrap:

- A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).
- A002468 The game of Mousetrap with n cards.
- A002469 The game of Mousetrap with n cards.
- A028305 Triangle of numbers of permutations eliminating just k cards out of n in game of Mousetrap.
- A028306 Triangle of numbers of permutations eliminating just card k out of n in game of Mousetrap.
- A018931 The number of permutations of n cards in which 2 is the first card hit and 3 the next hit after 2.
- A018932 The number of permutations of n cards in which 4 will be the next hit after 2.

Sequences related to SET:

- A182240 Number of inequivalent ways to select n cards in the game of SET.
- A090245 Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game.

Sequences related to Skat:

- A027887 Possible hands in the card game Skat.
- A027888 Possible hands in the card game Skat (abbreviated version).

Klondike solitaire: A071761, A214623, A214624.

Other card-related sequences:

- A161136 Triangular array S(m,n), 1<=n<=m, giving the total number of tossed cards in the card game described in A161135.
- A161135 Triangular array T(m,n), 1<=n<=m, giving the minimum positive number of deals of m cards into n piles required to collect all cards in the first pile. Each deal tosses all cards from a pile, the last dealt card indicates a pile to deal next, each deal tosses one card consecutively to the first, 2nd, ..., n-th, first, 2nd, ... pile.
- A061924 Number of combinations in card games with 4 suits and 4 players.
- A110017 Number of games that can be played with a set of 4n cards.
- A071761 Dealing cards in a game of solitaire.
- A076121 Complete list of possible cribbage hands.
- A143381 Number of Hi-Lo arrangements HL(m,n) of a deck with n suits and m ranks in each suit, m >= 1, n >= 1.

See also A225521.

###### Poker

poker See also 'poker' in the Index.

##### Mancala games

Mancala games involve seeds (or stones or other markers) in pits in a special board. The most common

- A002491 Smallest number of stones in Tchoukaillon (or Mancala, or Kalahari) solitaire that make use of n-th hole.
- A112557 Smallest number of stones in Tchoukaillon (or Mancala, or Kalahari) solitaire which make use of (2*n-1)-th hole for n>=1; a bisection of A002491.
- A028920 Pit harvesting sequence for winning solitaire Tchoukaillon (or Mancala).
- A007952 Tchoukaillon sieve
- A140060, A204539, A141262, A141271, A141272, A000201, A000960, A082447

Generalized Mancala solitaire: A112558, A113742, A113743, A113744, A113745, A113746, A113747, A113748, A113749, A113750

- A007780: Losing initial configurations in 2-hole Tchuka Ruma.

##### Heap games

Nim is the quintessential heap game, which has its own section in the Index. It is closely related to the Sprague–Grundy theorem of combinatorial game theory.

Bulgarian solitaire: A037481, A047996, A071762, A074909, A123975, A185700, A188160, A225794, A226062, A227141, A227147, A227451, A227452, A227453, A242422, A242424, A243051, A243052, A243053, A243060, A243070, A243072, A243073, A243353

Montreal Solitaire: A007075, A007046, A007048, A007049, A007050, A007076

##### Dice games

Sequences relating to Boggle:

- A063000 Number of paths of length n+2 originating at a corner of 4 X 4 Boggle board.
- A063001 Number of paths of length n+2 originating at a non-corner edge of 4 X 4 Boggle board.
- A063002 Number of paths of length n+2 originating at an interior vertex of 4 X 4 Boggle board.
- A111214 Score for an n-letter word in the game of Boggle.
- A236690 Number (assuming each character is unique) of strings that are possible in an n X n Boggle grid.

Sequences relating to backgammon:

- A025489 Numbers on backgammon doubling cube.
- A055100 In the game of backgammon, the number of blot hitting rolls when the blot is n points away, n=1,2,...,24.

##### Domino and tile games

Figures built out of dominoes: A056786, A216598, A216583, A216595, A216492, A216581

Number of chains: A284287

Game of dominoes: A031940, A008967, A045430, A258064

Packing a box with dominoes (or tilings): A001224, A004003, A006125, A001835, A002414, A003697, A003729, A003735, A003741, A003747, A003757, A003763, A003769, A003775, A004253, A005178, A007762, A028420, A038758, A054344, A001045

The five domino sequences: A108376, A108377, A108378, A108379, A108392

Luminations: A080546, A080547, A080548, A080549, A080550, A080551, A080552, A080553, A080554, A080555, A080556, A080557, A080558

Orbix: A010928, A080560, A080561, A080562, A080563, A080564

##### Role-playing games

Platonic solids are used in many role-playing games as dice (though sometimes others, like barrel-shaped dice, are used as well). A053016 lists the number of sides of the Platonic solids in 3D, and A060852 the sums of their pips. Perhaps a 4-dimensional role-playing game would use dice with A063924 cells.

- A124437 Experience Points thresholds for levels in the P&P-RPG "Das Schwarze Auge" (DSA, aka "The Dark Eye").

##### Other tabletop games

Tic-tac-toe is highly mathematical; the Hales–Jewett theorem proves that for a given number of players and board size, there is a *D* such that *d*-dimensional *D × D × ... × D* tic-tac-toe has a unique winner (and thus, by a strategy-stealing argument, the first player can force a win). Indeed, the stronger *density Hales–Jewett theorem* shows that an arbitrarily small fraction of the board need be filled before some player wins, if dimension is sufficiently large. See 'tic-tac-toe' in the Index.

Connect Four is very mathematical; it was game-theoretically solved by James D. Allen and independently by Victor Allis in 1988. Related sequences include A090224, A212693, and A013582.

For the Rubik's cube, see 'Rubik' in the Index.

The game of Blokus is directly related to polyominoes.

- A000105 The original game of Blokus uses all polyominoes of sizes 1 through 5.
- A000577 Blokus Trigon uses the polyiamonds of sizes 1 through 6.

Morpion solitaire: A204107, A204108, A204109, A204110

Triangular peg solitaire: A112737, A112738, A126777, A130515, A130516, A120422, A127500

Snooker: A128310, A180158, A241263

Sudoku: A107739, A109252, A109741, A112454, A114288, A140676, A159299, A173103?, A173104?, A175547, A182866, A185141, A198297; see also A211172, A216196, A229792, A269526, A273138, A274315, A274316, A274317, A274318, A274529, A274530, A274534, A274640, and A274791

#### Video games

Tetris uses one-sided tetrominos, giving a total of A000988(4) = 7 pieces. The sequences gives the number of one-sided polyominoes with a given number of blocks (4 in this case). A000105, A001419, A000104, A001168 are related sequences, which might be thought of as giving rise to different Tetris-like games. For example, A000105 corresponds to the variant allowing reflection of pieces in addition to rotation. A174248 gives the number of tilings of a 4 X n rectangle with n tetrominoes of any shape. A230031 is the number A(n,k) of tilings of a k X n rectangle using tetrominoes of any shape.

- A061418 Number of animals starting from a single pair if any pair of animals can produce a single offspring (as in the game Minecraft, if the player allows offspring to fully grow before breeding again).
- A119567 Lenny Conundrum #151: An apparently arbitrary list of numbers a(n) paired with 3*a(n) + 1.
- A119568 Lenny Conundrum #168: Neopet species in alphabetical order, converted to digits by the phone keypad code.
- A206344 Number of (potentially unsolvable) "clock puzzles" with
*n*positions in the video game Final Fantasy XIII-2. - A258231 Number of new Pokémon introduced in generation
*n*. - A259233 is the random number table for
*Doom*.

#### Casino games

- A100670 Number of two-card Baccarat hands of point n.
- A069892 Sequence around outside of single zero roulette wheel.
- A069893 Sequence around outside of double zero roulette wheel.
- A255821 Numbers of words on {0,1,...,36} having no isolated zeros. (Related to roulette probabilities.)
- A291154 Red numbers on the roulette wheel.
- A291171 Black numbers on the roulette wheel.

### Hobbies

#### Numismatics and notaphily

- A050947 Values of Russian money (in kopecks; 100 kopecks = 1 rouble).
- A070730 Euro coinage (in cents) from smallest physical size to largest.
- A080104 appeared on the Swiss 10 franc note.
- A112024 Number of cents in U.S. currency in 2005.
- A124146 U.S.A. currency denominations in dollars.
- A216606 discusses the seven-sided coins of the world.

#### Origami

See 'paper folding' in the Index.

#### Philately

## Athletics and sports

### Olympics

- A053402 Running distances (in meters) in Olympic athletics.

### Sports

- A009005 Along with 0, possible scores in rugby.
- A038717 Number of ways your team can score
*m*points in*n*rounds of a soccer competition (loss=0 point, draw=1 point, win=3 points). - A111872 a(n) is the number of runs Sir Donald Bradman scored in his n-th innings in test cricket.
- A167210 Tennis scores: values of successive points in a game.
- A187353 Number of Possible Ways to Fill Out NCAA Division I Men's Basketball Tournament Office Pool by Year, 1939-2011
- A244998 Number of ways for five teams of a World Cup football group to each have n goals for and n goals against.

#### American football

- A020725 Possible sums of the final scores of completed American football games. 1 point only is an impossible score in American football.
- A069745 Number of ways for a team to score n points in American football, where, except as noted below, order of scoring does not matter.
- A095973 Yard markers on a U.S.A. football field.
- A114143 The possible sums of the final scores of completed American football games where both teams score.
- A137684 Number of tie-less (American) football games with n scoring events.
- A157319 Possible total points for a single team in a game of American football, ignoring safeties (and time constraints).
- A160993 The number of ordered ways to achieve a score of n in American football.
- A237997 Number of ordered ways to achieve a score of n in American football taking into account different scoring methods.

#### Baseball

- A033545 Uniform numbers retired by New York Yankees.
- A033546 Uniform numbers retired by New York Yankees (in chronological order).
- A115380 Successive records for the number of home runs scored by a single batter in a single season of (USA) major league baseball.
- A121379 Successive records for the number of hits by a single batter in a single season of (USA) major league baseball.
- A121403 Decimal expansion of the area of home plate (USA major league baseball) in square inches.
- A136407 Valid strings, in lexicographic order, of Balls ("1") and Strikes ("2") in a Baseball at-bat. Numbers that contain only 1's and 2's never exceeding 3 total 2's or 4 total 1's, whichever comes first.
- A146293 Number of games in Major League Baseball's World Series for the year 1900 + n.
- A160175 Number of ways two opposing baseball teams could score a combined total of n runs (tallying the score just prior to each "batter up!") considering the order of the scoring as important.

#### Bowling

- A060853 Number of possible games of 10-pin bowling with a total score of
*n*. - A079596 Number of ways to get ten-pin bowling score of 300 -
*n*. - A127993 Minimum bowling score for a game with
*n*strikes. - A127994 Maximum bowling score for a game with exactly
*n*strikes.

#### Darts

- A008575 Sectors on darts board.
- A003833 Sectors around outside of darts board.
- A241746 Smallest number greater than n that CANNOT be scored using n darts on a standard dartboard.
- A242717 The number of ways that a score of n can be obtained using four darts on a standard dartboard.
- A242718 The number of ways that a score of n can be obtained using one dart on a standard dartboard.
- A242678 The number of ways that a score of n can be obtained using three darts on a standard dartboard.
- A242681 The number of ways that a score of n can be obtained using two darts on a standard dartboard.
- A020993 List of scores that can be achieved with four darts all of which hit a dartboard with regions labeled 1, 5, 10, 25.
- A076119 Every second sector of a dart board, starting at the top (20) and working around clockwise.
- A244512 The angle of the center of each numbered sector of a standard competition dart board.
- A117883 Alternate numbers on a dart board.
- A167213 The number of ordered ways to achieve a score of n in darts.
- A244196 Cumulative angle at the center of successive numbered sectors of a standard competition dart board, with numbers traversed in order.
- A104159 Numbers on a Manchester or Log-End dartboard, as read in a standard, clockwise direction.
- A233820 Sectors around outside of London Fives board.

#### International rules football

- A064422 Football league numbers: the possible point series for a league of n teams playing each other twice where for each match 3 points are awarded to the winning team and 1 to each in the case of a tie.
- A064626 Football tournament numbers: the number of possible point series for a tournament of n teams playing each other once where 3 points are awarded to the winning team and 1 to each in the case of a tie.
- A152789 Football tournament numbers with distinct point totals: number of point series in A064626 in which no two teams have the same total number of points.
- A209467 Football league numbers with distinct point totals for a league of n teams playing each other twice where for each match 3 points are awarded to the winning team and 1 to each in the case of a tie.

## Media

### Radio

- A056064 The Kubelsky sequence: Jack Benny's reported age, sampled annually.

### Television

- A027440 TV channels in Tokyo.
- A056064 The Kubelsky sequence: Jack Benny's reported age, sampled annually.
- A085808 Price is Right wheel.
- A104101 The Lost Numbers, a mysterious sequence of numbers appearing on the show
*Lost*. These numbers appeared on an episode of*Jeopardy!*as a red herring for the Fibonacci sequence: the $2000 clue in the Scientific European category, December 10, 2013, as "4, 8, 15, 16, 23, 42...wait, those aren't numbers in this guy's "sequence"! 3, 5, 8, 13, 21, 34... there we go." - A121313 Decimal expansion of 22π + 4
*e*. (John F. Nash mentioned this number in a postcard that was shown on the television program "The American Experience: A Beautiful Madness." He called it the "Beautiful Buddhist Number.") - A225579 The Amelia Code.
- A229093 was inspired by the Japanese drama 月の恋人 (
*Tsuki no Koibito*). - A229381 The Simpsons' perfect number, Mersenne prime, and narcissistic number.

#### Sequences appearing on television

The OEIS, and specifically A001608 and A059756, appear in Mr. Robot series 2 episode 11.

- A035497 Happy primes: primes that eventually reach 1 under iteration of "x -> sum of squares of digits of x". (Appeared as a security question in a Dr. Who episode.)
- A172984 Sequence congruent to the Fibonacci sequence modulo 5, with 1 added to the last term. Seen on "Mathnet" (This sequence was used as a puzzle in the "Mathnet" portion of the children's mathematics television show Square One TV.)

### Movies/Films

- A000014 is featured in
*Good Will Hunting*, in which the title character illustrates a(10), the number of "homeomorphically irreducible trees with n = 10." - A180028 and A180032 relate to the film
*Alice in Wonderland* - A246651 Red Squadron fighter numbers, in reported order, from the 1977 motion picture Star Wars.

### Internet

- A large number of integer sequences have been featured, directly or indirectly, on Numberphile: A023108, A000170, A134808, A016885, A000014, A003401, A006972, A094133, A016189, A138148, A036057, A129868, A146025, A048987, A011533, A060464, A080035, A164937, A001230, A105975, A008952, A212693, A260482, A260747, A260748, A260749, A260750, A007902, A064810, A218631, A242412, A057896, A198297, A230582, A265383, A216928, A050289, A007540, A006890, A051021, A105870, A144688, A014715, A080992, A065003, A001462, A002210; in fact, one sequence (A247698, the "Brady numbers") inspired by Numberphile.

Aperiodical has reviewed (humorously) a number of integer sequences.

## Chronology

### Calendars

#### Gregorian calendar

- A008684 Dates of successive days in Gregorian calendar.
- A008685 Lengths of months in the Gregorian calendar.
- A011763 Days in year in Gregorian calendar.
- A097105 Gregorian years containing "blue" Islamic New Year Days.
- A116369 Day of the week corresponding to Jan 01 of a given year (n=0 for the year 2000)
- A130447 Numbering the days of a 365-day year from 1 (Jan 01) to 365 (Dec 31), these are the days that start months.
- A184978 The months of the Gregorian calendar of less than 31 days.

Sequences related to *computus*, the calculation of the dates of Easter:

- A060958 Gregorian Easter dates, starting after 2000-04-23, a(2000) = 54.
- A084427 Gregorian calendar years with Ascension Day in April.
- A104019 Years in the Gregorian calendar for which Easter falls on the 25th day of the month.
- A104034 Years between occurrences of Easter on the 25th of the month in the Gregorian calendar.
- A134691 Possible number of weeks between two (Gregorian calendar) Easters.
- A156743 Month (3=March, 4=April) in which Easter occurs in Gregorian calendar year, starting with 1901.
- A224110 Easter occurrences on March 22, March 23, ..., April 25 during a 5,700,000-year Gregorian Easter cycle.
- A243001 a(n) = date of Easter in March of n, in the Gregorian calendar after A.D. 1582. a(n) = 0 if Easter occurs in April.
- A243002 a(n) = date of Easter in April of n, in the Gregorian calendar after A.D. 1582. a(n) = 0 if Easter occurs in March.

#### Mayan calendar

- A081244 Named periods in the Mayan/mesoamerican calendars.
- A081245 Number of days in months in the Haab year of Mayan/mesoamerican calendars.
- A115100 Mayan calendar periods in days.

#### Hebrew calendar

- A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra-month.
- A057350 Days in months in the Hebrew calendar starting from Nisan 5760 (Spring 2000 CE).

#### Islamic calendar

- A057347 Leap years in the Islamic calendar starting year 1 AH (Anno Hegirae) = 622 CE (Common Era or AD). There are 11 leap years in a 30 year cycle.
- A057348 Days in months in the Islamic calendar starting from Muharram, 1 AH. The twelfth month has 30 days in a leap year.
- A097105 Gregorian years containing "blue" Islamic New Year Days.

## Languages

### Alphabets, syllabaries, ideograms and pictograms

### Numerals

- A098378 Number of characters needed to write number n in the traditional Ethiopic (Geez) number system.

## Literature

### Cartoons

### Essays

### Novels

- A054382 is known as James Joyce's "Ulysses" sequence, for the novel's reference to the 369693100-digit number

which Joyce describes as requiring "innumerable quires and reams of India paper" to write down.9 9 9 - Scarlett Thomas' "The End of Mr Y" quotes (an uncorrected version of) A100200 in Chapter 12.
- A191299 lists the integers in Pynchon's "Gravity's Rainbow".
- A256590, the Base-2 Reacher numbers, are named for the character Jack Reacher in Lee Child's novel
*Bad Luck and Trouble*(among others in the series). - Verghese's novel "Cutting for Stone" references A047841(2) = 10213223 as "the only number that describes itself when you read it."

### Poems

- James Henle gives the terms of A005151 as an example of a mathematical poem.
^{[3]} - Inger Christensen's
*alphabet*has a collection of 14 poems, each with A000045(*n*+1) lines on the*n*-th letter of the alphabet. Brian Bilston's Word Crunching uses A000045 as the number of words in each line. - Pablo Picasso's "Poème: Mathématiquement pure image illusoire du ronflement écoeurant ..." contains a number sequence, A261903. Its meaning is unclear.
- A054639, the Queneau numbers, were inspired by the lyric structure of 12th century Occitan troubadour Arnaut Daniel's poem
*Lo ferm voler qu'el cor m'intra*and studied by novelist Raymond Queneau, founder of the Oulipo school of constrained writing. - The Bell numbers A000110 count the number of distinct rhyme schemes for an
*n*-line poem.

### Short stories

Andreas Bøe's *The Toy Robot* references the OEIS in a sub-chapter titled "Sloane".

## Philosophy

## Religion and theology

- A113515 Numbers of [Catholic] popes.
- A175726 Age of Biblical generations from Adam to Noah according to the Hebrew Bible.
- A248688 Number of chapters in the n-th book of the King James Version of the Holy Bible.
- A248689 Number of verses in the n-th book of the King James Version of the Holy Bible.
- A248690 Number of words in the n-th book of the King James Version of the Holy Bible.
- A248691 Number of letters in the n-th book of the King James Version of the Holy Bible.
- A255818 A method found in the Kol Bo (Hebrew: כלבו), a book of religious Jewish law from the 13th to 14th centuries, for calculating a person's age without anyone saying it explicitly.

## Arts

### Visual arts

#### Typography

- A002963 Chisel strokes required for Roman numerals for n.
- A002964 Smallest number requiring n chisel strokes for its representation in Roman numerals.
- A048181 Chisel strokes in Runes numerals (= alphabet characters).
- A051791 Numbers that are unchanged when turned upside down, when written in a font in which 7 looks like upside-down 2).
- A061745 Unicode codes for the Han digits.
- A098476 Unicode codes for the lunation runes, used in certain medieval Scandinavian perpetual calendar staves as golden numbers 1-19.
- A133192 Numbers that require exactly five chisel strokes when written in Roman numerals.
- A133194 Numbers requiring a prime number of chisel strokes for their representation in Roman numerals.
- A163670 Numbers whose English name (excluding spaces and hyphens) is written with only straight line segments (chisel strokes).
- A163828 Number of straight line segments in all letters of the capitalized English name of n.
- A164052 Numbers divisible by the number of chisel strokes in their English name (capitalized, excluding hyphens, excluding curves).
- A164059 Number of straight plus curved segments in the capitalized English name of n.
- A181698 Font point sizes under Mac OS Classic.
- A181699 Font point sizes under Mac OS X.
- A185374 Number of holes in n-th lower case letter of English alphabet (in printed form using standard Times New Roman font).
- A192741 Number of straight line segments in all letters of the capitalized Spanish name of n. Spanish version of A163828.
- A240564 A number (where A=1, B=2...) for every letter in the capitalized alphabet that does not have a curved line in the letter.
- A249574 The number of strokes for each capital letter of the alphabet from A to Z in Times New Roman font.

#### Calligraphy

#### Illustration

- A167600 7 X 7 square read by rows, from a silver gelatin print ("Diagram") by the American conceptual artist Mel Bochner.
- A245837 (0,1)-sequence forming an 10 X 14 rectangle, read by rows, from an illustration ("Barcode Flag") by Christian Tönsmann, art director of Süddeutsche Zeitung.

#### Photography

- A066844 Film speeds.

#### Cinematography

#### Plastic arts

##### Woodcuts

A number of OEIS sequences relate to counting the angels and devils in Escher's *Circle Limit IV*:^{[4]} A004396, A234271, A234272, A234273, A234274, A234275.

Albrecht Dürer's *Melencolia I* contains a magic square, the entries of which are A080992.

##### Sculpture

##### Architecture

Le Corbusier speculated about the proportions of an ideal man, in terms of simple quadratic irrationals such as the Golden ratio . He applied this ratio (and its reciprocal?) to his work. The proportions are given in A080104, A080105, A080106, A080078, and A080074.

Many of the proportions of the Parthenon (temple on the Athenian Acropolis, Greece, dedicated to the Greek goddess Athena, whom the people of Athens considered their virgin patron) (are alleged to?) exhibit the golden ratio.

A147704 links to a paper by Buitrago & Huylebrouck, titled "Nonagons in the Hagia Sophia and the Selimiye Mosque", which discusses the architecture of Mimar Sinan.

### Decorative arts

#### Textile arts

A193138 Number of square satins of order n (relates to satins and twills)

### Performing arts

#### Acrobatics

##### Circus

##### Juggling

Juggling, or more specifically toss juggling, is an art form that involves physically permuting a set of objects (generically called *props*, usually balls, clubs, rings, etc.) between the hands of performer(s), usually with an even tempo. The notation system for juggling patterns, called siteswap notation has a strong connection with permutations.

For sequences related to juggling, for a moment, please the section in the index: Index to OEIS: Section J#Juggling.

#### Theater

- A054356 The Five Hysterical Girls Theorem, created and mentioned in the play "The Five Hysterical Girls Theorem".

#### Dance

- A079908 - A079920 Solution to the Dancing School Problem with [3 girls and n+3 boys: f(3,n).] ... [15 girls and n+15 boys: f(15,n).]
- A079921 - A079928 Solution to the Dancing School Problem with n girls and [n+2 boys: f(n,2).] ... [n+9 boys: f(n,9).]

#### Music

*See also the OEIS index entry for music and Listen to a sequence*

The Pythagoreans developed a system of musical tuning based on the harmonic numbers. A different system was independently developed by Bharata. Many systems were later developed, such as those of Aron, Werckmeister, Vallotti, Young, Helmholtz, and Secor. The use of equal temperament was revived by Vincenzo Galilei. With such longstanding practical and music-theoretic interest in these systems it is no surprise that the OEIS contains many related sequences: A131062, A131071, A103922, A079731, A206788, A061920, A054540, A060525, A060526, A060527, A060528, A060529, A060233, A061918, A061919, A061416, A116474, A116475, A117577, A117578, A101286, A071831, A071832, A071833, A101285, A061921, A101287, A143800, A052307, A005664, A005663, A112732, A162918, A221363.

The 12th root of 2 figures in our standard 12-tone tuning of music today, where it represents the frequency ratio of a semitone in equal temperament (the equal-tempered chromatic scale divides the octave, which has a ratio of 2:1, into twelve equal parts).

- A010774 Decimal expansion of 12th root of 2.

Change ringing (usually of church bells) is highly mathematical, related to the combinatorics of permutations. Search "change ringing" to find examples of related sequences.

In regards to music, the OEIS also touches upon musicology and music history, listing for example the opus numbers of Beethoven's nine Symphonies (A001491), his five piano concertos (A169728), the Koechel numbers of Mozart's string quartets (A064172 and A134769), and the opus numbers skipped over by Carl Nielsen (A113529).

Several songs are recorded in the OEIS, such as the *Star Wars* theme (A145330), Mary had a little lamb (A120441), and a proposed theme song (A124856). The number of items received in The Twelve Days of Christmas gives rise to several sequences such as A000292, A002620, and A003991. See 'songs' in the Index.

A round is a type of canon. A canon can be at any interval, but usually at the unison (1) or octave (8). The Goldberg Variations by Johann Sebastian Bach consist of an aria with 30 variations. Every third variation (those with a number that is a multiple of 3, see A008585) in the Goldberg Variations is a canon. Furthermore, Var. 3 is a canon at unison (1), Var. 6 is a canon at the second, Var. 9 is a canon at the third, and so on and so forth to Var. 30, which is a canon at the tenth.^{[5]}

A number of sequences are related to musical scales, e.g.

- A035495 Musical scales consisting of n notes.
- A101285 Rounded frequencies in Hertz of the notes of the C major music scale beginning at A.
- A101286 Rounded frequencies in Hz of the notes of the chromatic music scale beginning at A.
- A101287 Rounded frequencies in Hz of the notes of the pentatonic music scale beginning at F#.
- A166361 Scale degrees of the roots of chords in a traditional "twelve-bar blues" in Western music.

A215606 and A215605 relate to interval sequences in Schoenberg 12-tone rows.

Piano keyboard sequences: A059620, A079731, A060106, A060107, A081031, A081032, A214832, A254531.

- A111167 Numbers appearing in metronome marks of music pieces, denoting beats per minute to indicate the tempo of the piece.

##### Lyrics

Lyrics of famous songs consisting of or prominently including number sequences include

- A027884 Figaro's opening aria in
*Le Nozze di Figaro*by Mozart. - A027885 Leporello's Register-Aria from Mozart's 'Don Giovanni'.
- A038674 A finite series from the lyrics of La Farolera, a Latin American traditional children's song.
- A085735 Lyrics of "Aquarius", by Boards of Canada.
- A060858 "0303456", a popular song by Rafaela Carra.
- A064373 "60-60-842", a popular song by the B52's.
- A091978 A common Swedish drinking song.
- A096582 From the "100 Green Bottles" song.
- A104175 From the words to the song "Jenny's Letterbox" by Tommy Tutone.
- A109869 Day numbers in Chris Cagle's song "What a beautiful day".
- A111729 Historical progression of years from the song "In The Year 2525" by Denny Zager and Rick Evans.
- A114867 Ages mentioned in the hit song "100 years" by "Five for Fighting".
- A157989 The numbers of the jingle for a popular Ontario pizza chain's phone number. Sung as: "nine, six, seven, eleven, eleven, .... " Jingle first introduced in 1978 and has since become a pop culture item for Canadians.
- A191514 Lehrer's elementary sequence.
- A193739 The integers occurring in the song "One man went to mow".
- A228245 The integers occurring in the song "Ten green bottles".
- A274137 Numbers in the lyrics of Mikhail Scherbakov's song "Shest', sem', vosem'..." ("Six, seven, eight...").

Other sequences related to song lyrics:

- A182369 Decimal expansion of (7^(e - 1/e) - 9)*Pi^2, also known as Jenny's constant.
- A246917 Number of letters in n-th word of the Czech national anthem, Kde domov muj.
- A247655 Number of letters in n-th word of The Star Spangled Banner (version 1).
- A247656 Number of letters in n-th word of The Star Spangled Banner (version 2).

## Notes

- ↑ Brian Hayes, "A question of numbers",
*American Scientist***84**:1 (1996), pp. 10–14. - ↑ R. B. Campbell, The effect of inbreeding constraints and offspring distribution on time to the most recent common ancestor, Journal of Theoretical Biology 382 (2015), pp. 74-80.
- ↑ James Henle, Is (some) mathematics poetry?,
*Journal of Humanistic Mathematics***1**:1 (2011), pp. 94-100. - ↑ John Choi and Nicholas Pippenger, Counting the angels and devils in Escher's Circle Limit IV,
*Journal of Humanistic Mathematics***5**:2 (2015), pp. 51-59. - ↑ Kent Wheeler Kennan,
*Counterpoint*2nd Ed. Englewood Cliffs, New Jersey: Prentice-Hall (1972): 90—91

## Cite this page as

Alonso del Arte, Alvin Hoover Belt, Daniel Forgues, and Charles R Greathouse IV, *The multi-faceted reach of the OEIS*.— From the On-Line Encyclopedia of Integer Sequences® Wiki (OEIS® Wiki). [https://oeis.org/wiki/The_multi-faceted_reach_of_the_OEIS]