OFFSET

0,3

COMMENTS

a(n) is the number of balls in a triangular pyramid in which each edge contains n balls.

One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012).

Also (1/6)*(n^3 + 3*n^2 + 2*n) is the number of ways to color the vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3 + 2*x3 + 3*x1*x2)/6.

Also the convolution of the natural numbers with themselves. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001

Connected with the Eulerian numbers (1, 4, 1) via 1*a(n-2) + 4*a(n-1) + 1*a(n) = n^3. - Gottfried Helms, Apr 15 2002

a(n) is sum of all the possible products p*q where (p,q) are ordered pairs and p + q = n + 1. E.g., a(5) = 5 + 8 + 9 + 8 + 5 = 35. - Amarnath Murthy, May 29 2003

Number of labeled graphs on n+3 nodes that are triangles. - Jon Perry, Jun 14 2003

Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - Mike Zabrocki, Nov 05 2004

Schlaefli symbol for this polyhedron: {3,3}.

Transform of n^2 under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

a(n) is a perfect square only for n = {1, 2, 48}. E.g., a(48) = 19600 = 140^2. - Alexander Adamchuk, Nov 24 2006

a(n+1) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4)^n. - Sergio Falcon, Feb 12 2007 [Corrected by Graeme McRae, Aug 28 2007]

a(n+1) is the number of terms in the complete homogeneous symmetric polynomial of degree n in 3 variables. - Richard Barnes, Sep 06 2017

This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007

a(n) = (d/dx)(S(n, x), x)|_{x = 2}. First derivative of Chebyshev S-polynomials evaluated at x = 2. See A049310. - Wolfdieter Lang, Apr 04 2007

If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007

Equals row sums of triangle A152205. - Gary W. Adamson, Nov 29 2008

a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12) = 364, almost the number of days in the year. - Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008

Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - Johannes W. Meijer, Mar 07 2009

Starting with 1 = row sums of triangle A158823. - Gary W. Adamson, Mar 28 2009

Wiener index of the path with n edges. - Eric W. Weisstein, Apr 30 2009

This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178: seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50). - Peter Luschny, Jul 14 2009

a(n) is the number of nondecreasing triples of numbers from a set of size n, and it is the number of strictly increasing triples of numbers from a set of size n+2. - Samuel Savitz, Sep 12 2009 [Corrected and enhanced by Markus Sigg, Sep 24 2023]

a(n) is the number of ordered sequences of 4 nonnegative integers that sum to n. E.g., a(2) = 10 because 2 = 2 + 0 + 0 + 0 = 1 + 1 + 0 + 0 = 0 + 2 + 0 + 0 = 1 + 0 + 1 + 0 = 0 + 1 + 1 + 0 = 0 + 0 + 2 + 0 = 1 + 0 + 0 + 1 = 0 + 1 + 0 + 1 = 0 + 0 + 1 + 1 = 0 + 0 + 0 + 2. - Artur Jasinski, Nov 30 2009

a(n) corresponds to the total number of steps to memorize n verses by the technique described in A173964. - Ibrahima Faye (ifaye2001(AT)yahoo.fr), Feb 22 2010

The number of (n+2)-bit numbers which contain two runs of 1's in their binary expansion. - Vladimir Shevelev, Jul 30 2010

a(n) is also, starting at the second term, the number of triangles formed in n-gons by intersecting diagonals with three diagonal endpoints (see the first column of the table in Sommars link). - Alexandre Wajnberg, Aug 21 2010

Column sums of:

1 4 9 16 25...

1 4 9...

1...

..............

--------------

1 4 10 20 35...

From Johannes W. Meijer, May 20 2011: (Start)

The Ca3, Ca4, Gi3 and Gi4 triangle sums (see A180662 for their definitions) of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Gi3(n) = 17*a(n) + 19*a(n-1) and Gi4(n) = 5*a(n) + a(n-1).

Furthermore the Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)

a(n-2)=N_0(n), n >= 1, with a(-1):=0, is the number of vertices of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - Wolfdieter Lang, May 27 2011

We consider optimal proper vertex colorings of a graph G. Assume that the labeling, i.e., coloring starts with 1. By optimality we mean that the maximum label used is the minimum of the maximum integer label used across all possible labelings of G. Let S=Sum of the differences |l(v) - l(u)|, the sum being over all edges uv of G and l(w) is the label associated with a vertex w of G. We say G admits unique labeling if all possible labelings of G is S-invariant and yields the same integer partition of S. With an offset this sequence gives the S-values for the complete graph on n vertices, n = 2, 3, ... . - K.V.Iyer, Jul 08 2011

Central term of commutator of transverse Virasoro operators in 4-D case for relativistic quantum open strings (ref. Zwiebach). - Tom Copeland, Sep 13 2011

Appears as a coefficient of a Sturm-Liouville operator in the Ovsienko reference on page 43. - Tom Copeland, Sep 13 2011

For n > 0: a(n) is the number of triples (u,v,w) with 1 <= u <= v <= w <= n, cf. A200737. - Reinhard Zumkeller, Nov 21 2011

Regarding the second comment above by Amarnath Murthy (May 29 2003), see A181118 which gives the sequence of ordered pairs. - L. Edson Jeffery, Dec 17 2011

The dimension of the space spanned by the 3-form v[ijk] that couples to M2-brane worldsheets wrapping 3-cycles inside tori (ref. Green, Miller, Vanhove eq. 3.9). - Stephen Crowley, Jan 05 2012

a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = n. Also, a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = 3n. - Clark Kimberling, Mar 19 2012

Using n + 4 consecutive triangular numbers t(1), t(2), ..., t(n+4), where n is the n-th term of this sequence, create a polygon by connecting points (t(1), t(2)) to (t(2), t(3)), (t(2), t(3)) to (t(3), t(4)), ..., (t(1), t(2)) to (t(n+3), t(n+4)). The area of this polygon will be one-half of each term in this sequence. - J. M. Bergot, May 05 2012

Pisano period lengths: 1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40, ... . (The Pisano sequence modulo m is the auxiliary sequence p(n) = a(n) mod m, n >= 1, for some m. p(n) is periodic for all sequences with rational g.f., like this one, and others. The lengths of the period of p(n) are quoted here for m>=1.) - R. J. Mathar, Aug 10 2012

a(n) is the maximum possible number of rooted triples consistent with any phylogenetic tree (level-0 phylogenetic network) containing exactly n+2 leaves. - Jesper Jansson, Sep 10 2012

For n > 0, the digital roots of this sequence A010888(a(n)) form the purely periodic 27-cycle {1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9, 9, 9}, which just rephrases the Pisano period length above. - Ant King, Oct 18 2012

a(n) is the number of functions f from {1, 2, 3} to {1, 2, ..., n + 4} such that f(1) + 1 < f(2) and f(2) + 1 < f(3). - Dennis P. Walsh, Nov 27 2012

a(n) is the Szeged index of the path graph with n+1 vertices; see the Diudea et al. reference, p. 155, Eq. (5.8). - Emeric Deutsch, Aug 01 2013

Also the number of permutations of length n that can be sorted by a single block transposition. - Vincent Vatter, Aug 21 2013

From J. M. Bergot, Sep 10 2013: (Start)

a(n) is the 3 X 3 matrix determinant

| C(n,1) C(n,2) C(n,3) |

| C(n+1,1) C(n+1,2) C(n+1,3) |

| C(n+2,1) C(n+2,2) C(n+2,3) |

(End)

In physics, a(n)/2 is the trace of the spin operator S_z^2 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and the sum of their squares is 10/2 = a(3)/2. - Stanislav Sykora, Nov 06 2013

a(n+1) = (n+1)*(n+2)*(n+3)/6 is also the dimension of the Hilbert space of homogeneous polynomials of degree n. - L. Edson Jeffery, Dec 12 2013

For n >= 4, a(n-3) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0111 (n-4 zeros), or, equivalently, a(n-3) is up-down coefficient {n,7} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014

a(n) is one-half the area of the region created by plotting the points (n^2,(n+1)^2). A line connects points (n^2,(n+1)^2) and ((n+1)^2, (n+2)^2) and a line is drawn from (0,1) to each increasing point. From (0,1) to (4,9) the area is 2; from (0,1) to (9,16) the area is 8; further areas are 20,40,70,...,2*a(n). - J. M. Bergot, May 29 2014

Beukers and Top prove that no tetrahedral number > 1 equals a square pyramidal number A000330. - Jonathan Sondow, Jun 21 2014

a(n+1) is for n >= 1 the number of nondecreasing n-letter words over the alphabet [4] = {1, 2, 3, 4} (or any other four distinct numbers). a(2+1) = 10 from the words 11, 22, 33, 44, 12, 13, 14, 23, 24, 34; which is also the maximal number of distinct elements in a symmetric 4 X 4 matrix. Inspired by the Jul 20 2014 comment by R. J. Cano on A000582. - Wolfdieter Lang, Jul 29 2014

Degree of the q-polynomial counting the orbits of plane partitions under the action of the symmetric group S3. Orbit-counting generating function is product_{i <= j <= k <= n} ( (1 - q^(i + j + k - 1))/(1 - q^(i + j + k - 2)) ). See q-TSPP reference. - Olivier Gérard, Feb 25 2015

If n is even then a(n) = Sum_{k=1..n/2} (2k)^2. If n is odd then a(n) = Sum_{k=0..(n-1)/2} (1+2k)^2. This can be illustrated as stacking boxes inside a square pyramid on plateaus of edge lengths 2k or 2k+1, respectively. The largest k are the 2k X 2k or (2k+1) X (2k+1) base. - R. K. Guy, Feb 26 2015

Draw n lines in general position in the plane. Any three define a triangle, so in all we see C(n,3) = a(n-2) triangles (6 lines produce 4 triangles, and so on). - Terry Stickels, Jul 21 2015

a(n-2) = fallfac(n,3)/3!, n >= 3, is also the number of independent components of an antisymmetric tensor of rank 3 and dimension n. Here fallfac is the falling factorial. - Wolfdieter Lang, Dec 10 2015

Number of compositions (ordered partitions) of n+3 into exactly 4 parts. - Juergen Will, Jan 02 2016

Number of weak compositions (ordered weak partitions) of n-1 into exactly 4 parts. - Juergen Will, Jan 02 2016

For n >= 2 gives the number of multiplications of two nonzero matrix elements in calculating the product of two upper n X n triangular matrices. - John M. Coffey, Jun 23 2016

Terms a(4n+1), n >= 0, are odd, all others are even. The 2-adic valuation of the subsequence of every other term, a(2n+1), n >= 0, yields the ruler sequence A007814. Sequence A275019 gives the 2-adic valuation of a(n). - M. F. Hasler, Dec 05 2016

Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 12 2017

C(n+2,3) is the number of ways to select 1 triple among n+2 objects, thus a(n) is the coefficient of x1^(n-1)*x3 in exponential Bell polynomial B_{n+2}(x1,x2,...), hence its link with A050534 and A001296 (see formula). - Cyril Damamme, Feb 26 2018

a(n) is also the number of 3-cycles in the (n+4)-path complement graph. - Eric W. Weisstein, Apr 11 2018

a(n) is the general number of all geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

a(n) + 4*a(n-1) + a(n-2) = n^3 = A000578(n), for n >= 0 (extending the a(n) formula given in the name). This is the Worpitzky identity for cubes. (Number of components of the decomposition of a rank 3 tensor in dimension n >= 1 into symmetric, mixed and antisymmetric parts). For a(n-2) see my Dec 10 2015 comment. - Wolfdieter Lang, Jul 16 2019

a(n) also gives the total number of regular triangles of length k (in some length unit), with k from {1, 2, ..., n}, in the matchstick arrangement with enclosing triangle of length n, but only triangles with the orientation of the enclosing triangle are counted. Row sums of unsigned A122432(n-1, k-1), for n >= 1. See the Andrew Howroyd comment in A085691. - Wolfdieter Lang, Apr 06 2020

a(n) is the number of bigrassmannian permutations on n+1 elements, i.e., permutations which have a unique left descent, and a unique right descent. - Rafael Mrden, Aug 21 2020

a(n-2) is the number of chiral pairs of colorings of the edges or vertices of a triangle using n or fewer colors. - Robert A. Russell, Oct 20 2020

a(n-2) is the number of subsets of {1,2,...,n} whose diameters are their size. For example, for n=4, a(2)=4 and the sets are {1,3}, {2,4}, {1,2,4}, {1,3,4}. - Enrique Navarrete, Dec 26 2020

For n>1, a(n-2) is the number of subsets of {1,2,...,n} in which the second largest element is the size of the subset. For example, for n=4, a(2)=4 and the sets are {2,3}, {2,4}, {1,3,4}, {2,3,4}. - Enrique Navarrete, Jan 02 2021

a(n) is the number of binary strings of length n+2 with exactly three 0's. - Enrique Navarrete, Jan 15 2021

From Tom Copeland, Jun 07 2021: (Start)

Aside from the zero, this sequence is the fourth diagonal of the Pascal matrix A007318 and the only nonvanishing diagonal (fourth) of the matrix representation IM = (A132440)^3/3! of the differential operator D^3/3!, when acting on the row vector of coefficients of an o.g.f., or power series.

M = e^{IM} is the lower triangular matrix of coefficients of the Appell polynomial sequence p_n(x) = e^{D^3/3!} x^n = e^{b. D} x^n = (b. + x)^n = Sum_{k=0..n} binomial(n,k) b_n x^{n-k}, where the (b.)^n = b_n have the e.g.f. e^{b.t} = e^{t^3/3!}, which is that for A025035 aerated with double zeros, the first column of M.

See A099174 and A000332 for analogous relationships for the third and fifth diagonals of the Pascal matrix. (End)

a(n) is the number of circles with a radius of integer length >= 1 and center at a grid point in an n X n grid. - Albert Swafford, Jun 11 2021

Maximum Wiener index over all connected graphs with n+1 vertices. - Allan Bickle, Jul 09 2022

The third Euler row (1,4,1) has an additional connection with the tetrahedral numbers besides the n^3 identity stated above: a^2(n) + 4*a^2(n+1) + a^2(n+2) = a(n^2+4n+4), which can be shown with algebra. E.g., a^2(2) + 4*a^2(3) + a^2(4) = 16 + 400 + 400 = a(16). Although an analogous thing happens with the (1,1) row of Euler's triangle and triangular numbers C(n+1,2) = A000217(n) = T(n), namely both T(n-1) + T(n) = n^2 and T^2(n-1) + T^2(n) = T(n^2) are true, only one (the usual identity) still holds for the Euler row (1,11,11,1) and the C(n,4) numbers in A000332. That is, the dot product of (1,11,11,1) with the squares of 4 consecutive terms of A000332 is not generally a term of A000332. - Richard Peterson, Aug 21 2022

For n > 1, a(n-2) is the number of solutions of the Diophantine equation x1 + x2 + x3 + x4 + x5 = n, subject to the constraints 0 <= x1, 1 <= x2, 2 <= x3, 0 <= x4 <= 1, 0 <= x5 and x5 is even. - Daniel Checa, Nov 03 2022

a(n+1) is also the number of vertices of the generalized Pitman-Stanley polytope with parameters 2, n, and vector (1,1, ... ,1), which is integrally equivalent to a flow polytope over the grid graph having 2 rows and n columns. - William T. Dugan, Sep 18 2023

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_0.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.

H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.

M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Nova Science, 2001, Huntington, N.Y. pp. 152-156.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics (no. 165), Cambridge Univ. Press, 2005.

Kenneth A Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 126-127.

B. Zwiebach, A First Course in String Theory, Cambridge, 2004; see p. 226.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.

Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 5.

F. Beukers and J. Top, On oranges and integral points on certain plane cubic curves, Nieuw Arch. Wiskd., IV (1988), Ser. 6, No. 3, 203-210.

Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

Gaston A. Brouwer, Jonathan Joe, Abby A. Noble, and Matt Noble, Problems on the Triangular Lattice, arXiv:2405.12321 [math.CO], 2024. Mentions this sequence.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

William Dowling and Nadia Lafreniere, Homomesy on permutations with toggling actions, arXiv:2312.02383 [math.CO], 2023. See page 8.

W. T. Dugan, M. Hegarty, A. H. Morales, and A. Raymond, Generalized Pitman-Stanley polytope: vertices and faces, arXiv:2307.09925 [math.CO], 2023.

Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020.

C. E. Frasser and G. N. Vostrov, Geodetic Graphs Homeomorphic to a Given Geodetic Graph, arXiv:1611.01873 [cs.DM], 2016. [p. 16, corollary 5]

Michael B. Green, Stephen D. Miller, and Pierre Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, arXiv:1111.2983 [hep-th], 2011-2013.

N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.

Jacob Hicks, M. A. Ollis, and John. R. Schmitt, Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches, arXiv:1809.02684 [math.CO], 2018.

A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 46. Book's website

Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.

C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946 [math.CO], 2013.

Milan Janjic, Two Enumerative Functions

Virginia Johnson and Charles K. Cook, Areas of Triangles and other Polygons with Vertices from Various Sequences, arXiv:1608.02420 [math.CO], 2016.

R. Jovanovic, First 2500 Tetrahedral numbers

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

M. Kobayashi, Enumeration of bigrassmannian permutations below a permutation in Bruhat order, arXiv:1005.3335 [math.CO], 2011; Order 28(1) (2011), 131-137.

C. Koutschan, M. Kauers, and D. Zeilberger, A Proof Of George Andrews' and David Robbins' q-TSPP Conjecture, Proc. Nat. Acad. Sc., vol. 108 no. 6 (2011), pp. 2196-2199. See also Zeilberger's comments on this article; Local copy of comments (pdf file).

T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.

P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - Juergen Will, Jan 02 2016

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See page 6.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (1).

Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Claude-Alexandre Simonetti, A new mathematical symbol : the termirial, arXiv:2005.00348 [math.GM], 2020.

N. J. A. Sloane, Illustration of initial terms

N. J. A. Sloane, Pyramid of 20 balls corresponding to a(3)=20.

S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.

H. Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals

G. Villemin's Almanach of Numbers, Nombres Tétraédriques (in French).

Eric Weisstein's World of Mathematics, Composition

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Path Complement Graph

Eric Weisstein's World of Mathematics, Path Graph

Eric Weisstein's World of Mathematics, Tetrahedral Number

Eric Weisstein's World of Mathematics, Wiener Index

Yue Zhang, Chunfang Zheng, and David Sankoff, Distinguishing successive ancient polyploidy levels based on genome-internal syntenic alignment, BMC Bioinformatics (2019) Vol. 20, 635.

A. F. Y. Zhao, Pattern Popularity in Multiply Restricted Permutations, Journal of Integer Sequences, 17 (2014), #14.10.3.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = C(n+2,3) = n*(n+1)*(n+2)/6 (see the name).

G.f.: x / (1 - x)^4.

a(n) = -a(-4 - n) for all in Z.

a(n) = Sum_{k=0..n} A000217(k) = Sum_{k=1..n} Sum_{j=0..k} j, partial sums of the triangular numbers.

a(n) = Sum_{1 <= i <= j <= n} |i - j|. - Amarnath Murthy, Aug 05 2002

a(n) = (n+3)*a(n-1)/n. - Ralf Stephan, Apr 26 2003

Sums of three consecutive terms give A006003. - Ralf Stephan, Apr 26 2003

Determinant of the n X n symmetric Pascal matrix M_(i, j) = C(i+j+2, i). - Benoit Cloitre, Aug 19 2003

The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005

a(n) = Sum_{k=0..floor((n-1)/2)} (n-2k)^2 [offset 0]; a(n+1) = Sum_{k=0..n} k^2*(1-(-1)^(n+k-1))/2 [offset 0]. - Paul Barry, Apr 16 2005

a(n) = -A110555(n+4, 3). - Reinhard Zumkeller, Jul 27 2005

Values of the Verlinde formula for SL_2, with g = 2: a(n) = Sum_{j=1..n-1} n/(2*sin^2(j*Pi/n)). - Simone Severini, Sep 25 2006

a(n-1) = (1/(1!*2!))*Sum_{1 <= x_1, x_2 <= n} |det V(x_1, x_2)| = (1/2)*Sum_{1 <= i,j <= n} |i-j|, where V(x_1, x_2) is the Vandermonde matrix of order 2. Column 2 of A133112. - Peter Bala, Sep 13 2007

Starting with 1 = binomial transform of [1, 3, 3, 1, ...]; e.g., a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson, Nov 04 2007

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Jaume Oliver Lafont, Nov 18 2008

Sum_{n>=1} 1/a(n) = 3/2, case x = 1 in Gradstein-Ryshik 1.513.7. - R. J. Mathar, Jan 27 2009

E.g.f.:((x^3)/6 + x^2 + x)*exp(x). - Geoffrey Critzer, Feb 21 2009

Limit_{n -> oo} A171973(n)/a(n) = sqrt(2)/2. - Reinhard Zumkeller, Jan 20 2010

With offset 1, a(n) = (1/6)*floor(n^5/(n^2 + 1)). - Gary Detlefs, Feb 14 2010

a(n) = Sum_{k = 1..n} k*(n-k+1). - Vladimir Shevelev, Jul 30 2010

a(n) = (3*n^2 + 6*n + 2)/(6*(h(n+2) - h(n-1))), n > 0, where h(n) is the n-th harmonic number. - Gary Detlefs, Jul 01 2011

a(n) = coefficient of x^2 in the Maclaurin expansion of 1 + 1/(x+1) + 1/(x+1)^2 + 1/(x+1)^3 + ... + 1/(x+1)^n. - Francesco Daddi, Aug 02 2011

a(n) = coefficient of x^4 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011

a(n) = 2*A002415(n+1)/(n+1). - Tom Copeland, Sep 13 2011

a(n) = A004006(n) - n - 1. - Reinhard Zumkeller, Mar 31 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1. - Ant King, Oct 18 2012

G.f.: x*U(0) where U(k) = 1 + 2*x*(k+2)/( 2*k+1 - x*(2*k+1)*(2*k+5)/(x*(2*k+5)+(2*k+2)/U(k+1) )); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Dec 01 2012

a(n^2 - 1) = (1/2)*(a(n^2 - n - 2) + a(n^2 + n - 2)) and

a(n^2 + n - 2) - a(n^2 - 1) = a(n-1)*(3*n^2 - 2) = 10*A024166(n-1), by Berselli's formula in A222716. - Jonathan Sondow, Mar 04 2013

G.f.: x + 4*x^2/(Q(0)-4*x) where Q(k) = 1 + k*(x+1) + 4*x - x*(k+1)*(k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013

a(n+1) = det(C(i+3,j+2), 1 <= i,j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013

a(n) = a(n-2) + n^2, for n > 1. - Ivan N. Ianakiev, Apr 16 2013

a(2n) = 4*(a(n-1) + a(n)), for n > 0. - Ivan N. Ianakiev, Apr 26 2013

G.f.: x*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

a(n) = n + 2*a(n-1) - a(n-2), with a(0) = a(-1) = 0. - Richard R. Forberg, Jul 11 2013

a(n)*(m+1)^3 + a(m)*(n+1) = a(n*m + n + m), for any nonnegative integers m and n. This is a 3D analog of Euler's theorem about triangular numbers, namely t(n)*(2m+1)^2 + t(m) = t(2nm + n + m), where t(n) is the n-th triangular number. - Ivan N. Ianakiev, Aug 20 2013

Sum_{n>=0} a(n)/(n+1)! = 2*e/3 = 1.8121878856393... . Sum_{n>=1} a(n)/n! = 13*e/6 = 5.88961062832... . - Richard R. Forberg, Dec 25 2013

Sum_{n>=1} (-1)^(n + 1)/a(n) = 12*log(2) - 15/2 = 0.8177661667... See A242024, A242023. - Richard R. Forberg, Aug 11 2014

3/(Sum_{n>=m} 1/a(n)) = A002378(m), for m > 0. - Richard R. Forberg, Aug 12 2014

a(n) = Sum_{i=1..n} Sum_{j=i..n} min(i,j). - Enrique Pérez Herrero, Dec 03 2014

Arithmetic mean of Square pyramidal number and Triangular number: a(n) = (A000330(n) + A000217(n))/2. - Luciano Ancora, Mar 14 2015

a(k*n) = a(k)*a(n) + 4*a(k-1)*a(n-1) + a(k-2)*a(n-2). - Robert Israel, Apr 20 2015

Dirichlet g.f.: (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1))/6. - Ilya Gutkovskiy, Jul 01 2016

a(n) = A080851(1,n-1) - R. J. Mathar, Jul 28 2016

a(n) = (A000578(n+1) - (n+1) ) / 6. - Zhandos Mambetaliyev, Nov 24 2016

G.f.: x/(1 - x)^4 = (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + x)^4 = (1 + 4x + 6x^2 + 4x^3 + x^4); and x/(1 - x)^4 = (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1 + x + x^2)^4. - Gary W. Adamson, Jan 23 2017

a(n) = Sum_{k=1..n} (-1)^(n-k)*A122432(n-1, k-1), for n >= 1, and a(0) = 0. - Wolfdieter Lang, Apr 06 2020

From Robert A. Russell, Oct 20 2020: (Start)

a(n) = 1*C(n,1) + 2*C(n,2) + 1*C(n,3), where the coefficient of C(n,k) is the number of unoriented triangle colorings using exactly k colors.

a(n-2) = 1*C(n,3), where the coefficient of C(n,k) is the number of chiral pairs of triangle colorings using exactly k colors.

a(n-2) = A327085(2,n). (End)

From Amiram Eldar, Jan 25 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = sinh(sqrt(2)*Pi)/(3*sqrt(2)*Pi).

Product_{n>=2} (1 - 1/a(n)) = sqrt(2)*sinh(sqrt(2)*Pi)/(33*Pi). (End)

EXAMPLE

a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.

Consider the square array

1 2 3 4 5 6 ...

2 4 6 8 10 12 ...

3 6 9 12 16 20 ...

4 8 12 16 20 24 ...

5 10 15 20 25 30 ...

...

then a(n) = sum of n-th antidiagonal. - Amarnath Murthy, Apr 06 2003

G.f. = x + 4*x^2 + 10*x^3 + 20*x^4 + 35*x^5 + 56*x^6 + 84*x^7 + 120*x^8 + 165*x^9 + ...

Example for a(3+1) = 20 nondecreasing 3-letter words over {1,2,3,4}: 111, 222, 333; 444, 112, 113, 114, 223, 224, 122, 224, 133, 233, 144, 244, 344; 123, 124, 134, 234. 4 + 4*3 + 4 = 20. - Wolfdieter Lang, Jul 29 2014

Example for a(4-2) = 4 independent components of a rank 3 antisymmetric tensor A of dimension 4: A(1,2,3), A(1,2,4), A(1,3,4) and A(2,3,4). - Wolfdieter Lang, Dec 10 2015

MAPLE

a:=n->n*(n+1)*(n+2)/6; seq(a(n), n=0..50);

isA000292 := proc(n)

option remember;

local a, i ;

for i from iroot(6*n, 3)-1 do

a := A000292(i) ;

if a > n then

return false;

elif a = n then

return true;

end if;

end do:

end proc: # R. J. Mathar, Aug 14 2024

MATHEMATICA

Table[Binomial[n + 2, 3], {n, 0, 20}] (* Zerinvary Lajos, Jan 31 2010 *)

Accumulate[Accumulate[Range[0, 50]]] (* Harvey P. Dale, Dec 10 2011 *)

Table[n (n + 1)(n + 2)/6, {n, 0, 100}] (* Wesley Ivan Hurt, Sep 25 2013 *)

Nest[Accumulate, Range[0, 50], 2] (* Harvey P. Dale, May 24 2017 *)

Binomial[Range[20] + 1, 3] (* Eric W. Weisstein, Sep 08 2017 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 4, 10}, 20] (* Eric W. Weisstein, Sep 08 2017 *)

CoefficientList[Series[x/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

Table[Range[n].Range[n, 1, -1], {n, 0, 50}] (* Harvey P. Dale, Mar 02 2024 *)

PROG

(PARI) a(n) = (n) * (n+1) * (n+2) / 6 \\ corrected by Harry J. Smith, Dec 22 2008

(PARI) a=vector(10000); a[2]=1; for(i=3, #a, a[i]=a[i-2]+i*i); \\ Stanislav Sykora, Nov 07 2013

(PARI) is(n)=my(k=sqrtnint(6*n, 3)); k*(k+1)*(k+2)==6*n \\ Charles R Greathouse IV, Dec 13 2016

(Haskell)

a000292 n = n * (n + 1) * (n + 2) `div` 6

a000292_list = scanl1 (+) a000217_list

-- Reinhard Zumkeller, Jun 16 2013, Feb 09 2012, Nov 21 2011

(Maxima) A000292(n):=n*(n+1)*(n+2)/6$ makelist(A000292(n), n, 0, 60); /* Martin Ettl, Oct 24 2012 */

(Magma) [n*(n+1)*(n+2)/6: n in [0..50]]; // Wesley Ivan Hurt, Jun 03 2014

(GAP) a:=n->Binomial(n+2, 3);; A000292:=List([0..50], n->a(n)); # Muniru A Asiru, Feb 28 2018

(Python) # Compare A000217.

def A000292():

x, y, z = 1, 1, 1

yield 0

while True:

yield x

x, y, z = x + y + z + 1, y + z + 1, z + 1

a = A000292(); print([next(a) for i in range(45)]) # Peter Luschny, Aug 03 2019

CROSSREFS

Sums of 2 consecutive terms give A000330.

Column 0 of triangle A094415.

Cf. A000217 (first differences), A001044, (see above example), A061552, A040977, A133111, A133112, A152205, A158823, A156925, A157703, A173964, A058187, A190717, A190718, A100440, A181118, A222716.

Partial sums are A000332. - Jonathan Vos Post, Mar 27 2011

Cf. A216499 (the analogous sequence for level-1 phylogenetic networks).

Cf. similar sequences listed in A237616.

Cf. A104712 (second column, if offset is 2).

Cf. A145397 (non-tetrahedral numbers). - Daniel Forgues, Apr 11 2015

Cf. A127324.

Cf. A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).

Cf. A002817 (4-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5})

Cf. (triangle colorings) A006527 (oriented), A000290 (achiral), A327085 (chiral simplex edges and ridges).

Row 3 of A321791 (cycles of n colors using k or fewer colors).

KEYWORD

nonn,core,easy,nice

AUTHOR

EXTENSIONS

Corrected and edited by Daniel Forgues, May 14 2010

STATUS

approved