0,5

Also number of ways to insert two pairs of parentheses into a string of n-4 letters (allowing empty pairs of parentheses). E.g., there are 30 ways for 2 letters. Cf. A002415.

2,10,30,70, ... gives orchard crossing number of complete graph K_n. - Ralf Stephan, Mar 28 2003

If Y is a 2-subset of an n-set X then, for n>=4, a(n-1) is the number of 4-subsets and 5-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007

Middle column of table on p. 6 of Feder and Garber. - Jonathan Vos Post, Apr 23 2009

Number of pairs of non-intersecting lines when each of n points around a circle is joined to every other point by straight lines. A pair of lines is considered non-intersecting if the lines do not intersect in either the interior or the boundary of a circle. - Melvin Peralta, Feb 05 2016

From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - Bruno Berselli, Oct 24 2016

Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

For n >= 3, also the number of tetrahedra in the n-barbell graph. - Eric W. Weisstein, Dec 19 2025

a(n) is the total number of runs of ones over the binary strings of length n-1 that contain exactly two runs of ones (see A000332). - Félix Balado, Sep 23 2025

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.

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

Bruno Berselli, Table of n, a(n) for n = 0..1000

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.

Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 105.

Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.

Elie Feder and David Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Eric Weisstein's World of Mathematics, Barbell Graph.

Eric Weisstein's World of Mathematics, Graph Cycle.

Eric Weisstein's World of Mathematics, Graph Tetrahedron.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - R. J. Mathar, Nov 08 2010

G.f.: 2*x^4/(1-x)^5. - Colin Barker, Feb 29 2012

a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - Wesley Ivan Hurt, Sep 26 2013

E.g.f.: x^4*exp(x)/12. - G. C. Greubel, Feb 23 2017

From Amiram Eldar, Jul 19 2022: (Start)

Sum_{n>=4} 1/a(n) = 2/3.

Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)

[seq(binomial(n, 4)*2, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* Harvey P. Dale, Jun 09 2016 *)

Table[2 Binomial[n, 4], {n, 0, 40}] (* Bruno Berselli, Oct 24 2016 *)

2 Binomial[Range[0, 20], 4] (* Eric W. Weisstein, Aug 10 2017 *)

(Magma) [2*Binomial(n, 4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013

(PARI) a(n)=2*binomial(n, 4) \\ Charles R Greathouse IV, Jun 23 2015

A diagonal of A088617.

Cf. A033487, A050534, A060008.

Partial sums of A007290.

Cf. A001477, A002378.

Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

Cf. A000332 (number of binary strings of length n-1 that contain exactly two runs of ones).

Sequence in context: A034262 A180499 A167214 * A328532 A051667 A106355

Adjacent sequences: A034824 A034825 A034826 * A034828 A034829 A034830

nonn,easy

approved