A000096 a(n) = n*(n+3)/2. (Formerly M1356 N0522)

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377, 1430, 1484, 1539, 1595, 1652, 1710, 1769

Offset

0,2

Comments

For n >= 1, a(n) is the maximal number of pieces that can be obtained by cutting an annulus with n cuts. See illustration. - Robert G. Wilson v

n(n-3)/2 (n >= 3) is the number of diagonals of an n-gon. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)

n(n-3)/2 (n >= 4) is the degree of the third-smallest irreducible presentation of the symmetric group S_n (cf. James and Kerber, Appendix 1).

a(n) is also the multiplicity of the eigenvalue (-2) of the triangle graph Delta(n+1). (See p. 19 in Biggs.) - Felix Goldberg (felixg(AT)tx.technion.ac.il), Nov 25 2001

For n > 3, a(n-3) = dimension of the traveling salesman polytope T(n). - Benoit Cloitre, Aug 18 2002

Also counts quasi-dominoes (quasi-2-ominoes) on an n X n board. Cf. A094170-A094172. - Jon Wild, May 07 2004

Coefficient of x^2 in (1 + x + 2*x^2)^n. - Michael Somos, May 26 2004

a(n) is the number of "prime" n-dimensional polyominoes. A "prime" n-polyomino cannot be formed by connecting any other n-polyominoes except for the n-monomino and the n-monomino is not prime. E.g., for n=1, the 1-monomino is the line of length 1 and the only "prime" 1-polyominoes are the lines of length 2 and 3. This refers to "free" n-dimensional polyominoes, i.e., that can be rotated along any axis. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005

Solutions to the quadratic equation q(m, r) = (-3 +- sqrt(9 + 8(m - r))) / 2, where m - r is included in a(n). Let t(m) = the triangular number (A000217) less than some number k and r = k - t(m). If k is neither prime nor a power of two and m - r is included in A000096, then m - q(m, r) will produce a value that shares a divisor with k. - Andrew S. Plewe, Jun 18 2005

Sum_{k=2..n+1} 4/(k*(k+1)*(k-1)) = ((n+3)*n)/((n+2)*(n+1)). Numerator(Sum_{k=2..n+1} 4/(k*(k+1)*(k-1)) = (n+3)*n/2. - Alexander Adamchuk, Apr 11 2006

Number of rooted trees with n+3 nodes of valence 1, no nodes of valence 2 and exactly two other nodes. I.e., number of planted trees with n+2 leaves and exactly two branch points. - Theo Johnson-Freyd (theojf(AT)berkeley.edu), Jun 10 2007

If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-2)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

For n >= 1, a(n) is the number of distinct shuffles of the identity permutation on n+1 letters with the identity permutation on 2 letters (12). - Camillia Smith Barnes, Oct 04 2008

If s(n) is a sequence defined as s(1) = x, s(n) = kn + s(n-1) + p for n > 1, then s(n) = a(n-1)*k + (n-1)*p + x. - Gary Detlefs, Mar 04 2010

The only primes are a(1) = 2 and a(2) = 5. - Reinhard Zumkeller, Jul 18 2011

a(n) = m such that the (m+1)-th triangular number minus the m-th triangular number is the (n+1)-th triangular number: (m+1)(m+2)/2 - m(m+1)/2 = (n+1)(n+2)/2. - Zak Seidov, Jan 22 2012

For n >= 1, number of different values that Sum_{k=1..n} c(k)*k can take where the c(k) are 0 or 1. - Joerg Arndt, Jun 24 2012

Sum_{n>0} 1/a(n) = 11/9. - Enrique Pérez Herrero, Nov 26 2013

On an n X n chessboard (n >= 2), the number of possible checkmate positions in the case of king and rook versus a lone king is 0, 16, 40, 72, 112, 160, 216, 280, 352, ..., which is 8*a(n-2). For a 4 X 4 board the number is 40. The number of positions possible was counted including all mirror images and rotations for all four sides of the board. - Jose Abutal, Nov 19 2013

If k = a(i-1) or k = a(i+1) and n = k + a(i), then C(n, k-1), C(n, k), C(n, k+1) are three consecutive binomial coefficients in arithmetic progression and these are all the solutions. There are no four consecutive binomial coefficients in arithmetic progression. - Michael Somos, Nov 11 2015

a(n-1) is also the number of independent components of a symmetric traceless tensor of rank 2 and dimension n >= 1. - Wolfdieter Lang, Dec 10 2015

Numbers k such that 8k + 9 is a square. - Juri-Stepan Gerasimov, Apr 05 2016

a(n) = (n+1)^2 - A000124(n).- Anton Zakharov, Jun 29 2016

Let phi_(D,rho) be the average value of a generic degree D monic polynomial f when evaluated at the roots of the rho-th derivative of f, expressed as a polynomial in the averaged symmetric polynomials in the roots of f. [See the Wojnar et al. link] The "last" term of phi_(D,rho) is a multiple of the product of all roots of f; the coefficient is expressible as a polynomial h_D(N) in N:=D-rho. These polynomials are of the form h_D(N)= ((-1)^D/(D-1)!)*(D-N)*N^chi*g_D(N) where chi = (1 if D is odd, 0 if D is even) and g_D(N) is a monic polynomial of degree (D-2-chi). Then a(n) are the negated coefficients of the next to the highest order term in the polynomials N^chi*g_D(N), starting at D=3. - Gregory Gerard Wojnar, Jul 19 2017

For n >= 2, a(n) is the number of summations required to solve the linear regression of n variables (n-1 independent variables and 1 dependent variable). - Felipe Pedraza-Oropeza, Dec 07 2017

For n >= 2, a(n) is the number of sums required to solve the linear regression of n variables: 5 for two variables (sums of X, Y, X^2, Y^2, X*Y), 9 for 3 variables (sums of X1, X2, Y1, X1^2, X1*X2, X1*Y, X2^2, X2*Y, Y^2), and so on. - Felipe Pedraza-Oropeza, Jan 11 2018

a(n) is the area of a triangle with vertices at (n, n+1), ((n+1)*(n+2)/2, (n+2)*(n+3)/2), ((n+2)^2, (n+3)^2)). - J. M. Bergot, Jan 25 2018

Number of terms less than 10^k: 1, 4, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, ... - Muniru A Asiru, Jan 25 2018

a(n) is also the number of irredundant sets in the (n+1)-path complement graph for n > 2. - Eric W. Weisstein, Apr 11 2018

a(n) is also the largest number k such that the largest Dyck path of the symmetric representation of sigma(k) has exactly n peaks, n >= 1. (Cf. A237593.) - Omar E. Pol, Sep 04 2018

For n > 0, a(n) is the number of facets of associahedra. Cf. A033282 and A126216 and their refinements A111785 and A133437 for related combinatorial and analytic constructs. See p. 40 of Hanson and Sha for a relation to projective spaces and string theory. - Tom Copeland, Jan 03 2021

For n > 0, a(n) is the number of bipartite graphs with 2n or 2n+1 edges, no isolated vertices, and a stable set of cardinality 2. - Christian Barrientos, Jun 13 2022

For n >= 2, a(n-2) is the number of permutations in S_n which are the product of two different transpositions of adjacent points. - Zbigniew Wojciechowski, Mar 31 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), Table 22.7, p. 797.

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.

G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Maths. and its Appls., Vol. 16, Addison-Wesley, 1981, Reading, MA, U.S.A.

D. G. Kendall et al., Shape and Shape Theory, Wiley, 1999; see p. 4.

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).

Links

Franklin T. Adams-Watters, 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].

D. Applegate, R. Bixby, V. Chvatal and W. Cook, On the solution of traveling salesman problem, In : Int. Congress of mathematics (Berlin 1998), Documenta Math., Extra Volume ICM 1998, Vol. III, pp. 645-656.

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.

L. Euler, Sur une contradiction apparente dans la doctrine des lignes courbes, Mémoires de l'Académie des Sciences de Berlin, 4, 219-233, 1750 Reprinted in Opera Omnia, Series I, Vol. 26. pp. 33-45.

Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018.

Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Theorem 1.

A. Hanson and J. Sha, A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6, arXiv preprint arXiv:0510064 [math-ph], 2005.

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.

S. P. Humphries, Home page

S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1018

Milan Janjic, Two Enumerative Functions

A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.

Yashar Memarian, On the Maximum Number of Vertices of Minimal Embedded Graphs, arXiv:0910.2469 [math.CO], 2009-15. [Jonathan Vos Post, Oct 14 2009]

Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.

E. Pérez Herrero, Binomial Matrix (I), Psychedelic Geometry Blogspot 09/22/09. [Enrique Pérez Herrero, Sep 22 2009]

T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.

P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.

P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Maria J. Rodriguez, Black holes in all, The KIAS Newsletter, Vol.3, pp.29-34, Korea Institute for Advanced Study, Dec. 2010.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

E. Sandifer, How Euler Did It: Cramer's Paradox

N. J. A. Sloane, How to cut an annulus into 9 pieces with three cuts.

Eric Weisstein's World of Mathematics, Cramer-Euler Paradox

Eric Weisstein's World of Mathematics, Irredundant Set

Eric Weisstein's World of Mathematics, Path Complement Graph

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Index entries for sequences related to Chebyshev polynomials.

Formula

G.f.: A(x) = x*(2-x)/(1-x)^3. a(n) = binomial(n+1, n-1) + binomial(n, n-1).

Connection with triangular numbers: a(n) = A000217(n+1) - 1.

a(n) = a(n-1) + n + 1. - Bryan Jacobs (bryanjj(AT)gmail.com), Apr 01 2005

a(n) = 2*t(n) - t(n-1) where t() are the triangular numbers, e.g., a(5) = 2*t(5) - t(4) = 2*15 - 10 = 20. - Jon Perry, Jul 23 2003

a(-3-n) = a(n). - Michael Somos, May 26 2004

2*a(n) = A008778(n) - A105163(n). - Creighton Dement, Apr 15 2005

a(n) = C(3+n, 2) - C(3+n, 1). - Zerinvary Lajos, Dec 09 2005

a(n) = A067550(n+1) / A067550(n). - Alexander Adamchuk, May 20 2006

a(n) = A126890(n,1) for n > 0. - Reinhard Zumkeller, Dec 30 2006

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2008

Starting (2, 5, 9, 14, ...) = binomial transform of (2, 3, 1, 0, 0, 0, ...). - Gary W. Adamson, Jul 03 2008

For n >= 0, a(n+2) = b(n+1) - b(n), where b(n) is the sequence A005586. - K.V.Iyer, Apr 27 2009

A002262(a(n)) = n. - Reinhard Zumkeller, May 20 2009

Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jul 08 2010

a(n) = Sum_{k=1..n} (k+1)!/k!. - Gary Detlefs, Aug 03 2010

a(n) = n(n+1)/2 + n = A000217(n) + n. - Zak Seidov, Jan 22 2012

E.g.f.: F(x) = 1/2*x*exp(x)*(x+4) satisfies the differential equation F''(x) - 2*F'(x) + F(x) = exp(x). - Peter Bala, Mar 14 2012

a(n) = binomial(n+3, 2) - (n+3). - Robert G. Wilson v, Mar 15 2012

a(n) = A181971(n+1, 2) for n > 0. - Reinhard Zumkeller, Jul 09 2012

a(n) = A214292(n+2, 1). - Reinhard Zumkeller, Jul 12 2012

G.f.: -U(0) where U(k) = 1 - 1/((1-x)^2 - x*(1-x)^4/(x*(1-x)^2 - 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 27 2012

A023532(a(n)) = 0. - Reinhard Zumkeller, Dec 04 2012

a(n) = A014132(n,n) for n > 0. - Reinhard Zumkeller, Dec 12 2012

a(n-1) = (1/n!)*Sum_{j=0..n} binomial(n,j)*(-1)^(n-j)*j^n*(j-1). - Vladimir Kruchinin, Jun 06 2013

a(n) = 2n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013

a(n) = Sum_{i=2..n+1} i. - Wesley Ivan Hurt, Jun 28 2013

a(n) = Sum_{i=1..n} (n - i + 2). - Wesley Ivan Hurt, Mar 31 2014

A023531(a(n)) = 1. - Reinhard Zumkeller, Feb 14 2015

For n > 0: a(n) = A101881(2*n-1). - Reinhard Zumkeller, Feb 20 2015

a(n) + a(n-1) = A008865(n+1) for all n in Z. - Michael Somos, Nov 11 2015

a(n+1) = A127672(4+n, n), n >= 0, where A127672 gives the coefficients of the Chebyshev C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015

a(n) = (n+1)^2 - A000124(n). - Anton Zakharov, Jun 29 2016

Dirichlet g.f.: (zeta(s-2) + 3*zeta(s-1))/2. - Ilya Gutkovskiy, Jun 30 2016

a(n) = 2*A000290(n+3) - 3*A000217(n+3). - J. M. Bergot, Apr 04 2018

a(n) = Stirling2(n+2, n+1) - 1. - Peter Luschny, Jan 05 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 5/9. - Amiram Eldar, Jan 10 2021

From Amiram Eldar, Jan 20 2021: (Start)

Product_{n>=1} (1 + 1/a(n)) = 3.

Product_{n>=1} (1 - 1/a(n)) = 3*cos(sqrt(17)*Pi/2)/(4*Pi). (End)

Example

G.f. = 2*x + 5*x^2 + 9*x^3 + 14*x^4 + 20*x^5 + 27*x^6 + 35*x^7 + 44*x^8 + 54*x^9 + ...

Maple

A000096 := n->n*(n+3)/2; seq(A000096(n), n=0..50);
A000096 :=z*(-2+z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

Table[n*(n+3)/2, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
LinearRecurrence[{3, -3, 1}, {0, 2, 5}, 60] (* Harvey P. Dale, Apr 30 2013 *)

Prog

(PARI) {a(n) = n * (n+3)/2}; \\ Michael Somos, May 26 2004
(PARI) first(n) = Vec(x*(2-x)/(1-x)^3 + O(x^n), -n) \\ Iain Fox, Dec 12 2017
(Haskell)
a000096 n = n * (n + 3) `div` 2
a000096_list = [x | x <- [0..], a023531 x == 1]
-- Reinhard Zumkeller, Feb 14 2015, Dec 04 2012
(Magma) [n*(n+3)/2: n in [0..60]]; or [n: n in [0..2300] | IsSquare(8*n+9)]; // Juri-Stepan Gerasimov, Apr 05 2016
(GAP) a := List([0..1000], n -> n*(n+3)/2); # Muniru A Asiru, Jan 25 2018

Crossrefs

Complement of A007401. Column 2 of A145324. Column of triangle A014473, first skew subdiagonal of A033282, a diagonal of A079508.

Occurs as a diagonal in A074079/A074080, i.e., A074079(n+3, n) = A000096(n-1) for all n >= 2. Also A074092(n) = 2^n * A000096(n-1) after n >= 2.

Cf. A000124, A000217, A005581-A005584, A023531, A034856, A067550, A127672 (Chebyshev C).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488.

Similar sequences are listed in A316466.

Cf. A033282, A111785, A126216, A133437.

Sequence in context: A212342 A080956 A132337 * A134189 A109470 A112873

Adjacent sequences: A000093 A000094 A000095 * A000097 A000098 A000099

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 04 2000

Status

approved