A027789 a(n) = 2*(n+1)*binomial(n+3,4).

4, 30, 120, 350, 840, 1764, 3360, 5940, 9900, 15730, 24024, 35490, 50960, 71400, 97920, 131784, 174420, 227430, 292600, 371910, 467544, 581900, 717600, 877500, 1064700, 1282554, 1534680, 1824970, 2157600, 2537040, 2968064, 3455760, 4005540, 4623150, 5314680

Offset: 1

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 8-subsequences of [ 1, n ] with just 3 contiguous pairs.

Also the number of 3-cycles in the n+3 tetrahedral graph. - Eric W. Weisstein, Jul 12 2017

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, Tetrahedral Graph.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A006470, A289792 (4-cycles), A289793 (5-cycles), A289794 (6-cycles).

Programs

Magma

[2*(n+1)*Binomial(n+3,4): n in [1..40]]; // Vincenzo Librandi, Jul 13 2017

Maple

A027789:=n->2*(n+1)*binomial(n+3,4): seq(A027789(n), n=1..60); # Wesley Ivan Hurt, Oct 23 2017

Mathematica

Table[2 (n + 1) Binomial[n + 3, 4], {n, 40}] (* Harvey P. Dale, Jan 20 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {4, 30, 120, 350, 840, 1764},40] (* Harvey P. Dale, Jan 20 2015 *)
Table[n (1 + n)^2 (2 + n) (3 + n)/12, {n, 20}] (* Eric W. Weisstein, Jul 12 2017 *)
CoefficientList[Series[(2 (2 + 3 x))/(-1 + x)^6, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 12 2017 *)

PARI

for(n=1,50, print1(2*(n+1)*binomial(n+3,4), ", ")) \\ G. C. Greubel, Oct 22 2017

Formula

G.f.: 2*(2+3x)*x/(1-x)^6.

a(n) = 2*A006470(n).

a(n) = C(n+1, 2)*C(n+3, 3). - Zerinvary Lajos, May 10 2005, corrected by R. J. Mathar, Feb 13 2016

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Jan 20 2015

a(n) = Sum_{k=1..n+1} Sum_{i=1..n+1} (n-i+1) * C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017

From Amiram Eldar, Jan 28 2022: (Start)

Sum_{n>=1} 1/a(n) = 61/6 - Pi^2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 8*log(2) + 5/6. (End)