A027790 a(n) = 10*(n+1)*binomial(n+3,5)/3.

10, 80, 350, 1120, 2940, 6720, 13860, 26400, 47190, 80080, 130130, 203840, 309400, 456960, 658920, 930240, 1288770, 1755600, 2355430, 3116960, 4073300, 5262400, 6727500, 8517600, 10687950, 13300560, 16424730, 20137600

Offset: 2

Author

thi ngoc dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 9-subsequences of [ 1, n ] with just 3 contiguous pairs.

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000292, A040977.

Programs

Magma

A027790:= func< n | 10*(n+1)*Binomial(n+3,5)/3 >;
[A027790(n): n in [2..40]]; // G. C. Greubel, Feb 21 2025

Mathematica

Table[10(n+1) Binomial[n+3,5]/3,{n,2,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{10,80,350,1120,2940,6720,13860},30] (* Harvey P. Dale, Jan 15 2015 *)

SageMath

def A027790(n): return 10*(n+1)*binomial(n+3,5)//3
print([A027790(n) for n in range(2,41)]) # G. C. Greubel, Feb 21 2025

Formula

G.f.: 10*(1+x)*x^2/(1-x)^7.

a(n) = binomial(n+1, 3)*binomial(n+3, 3) = A000292(n-1)*A000292(n+1). - Zerinvary Lajos, May 13 2005

a(n) = 10*A040977(n). - R. J. Mathar, May 22 2013

From Amiram Eldar, Jan 06 2021: (Start)

Sum_{n>=2} 1/a(n) = 3*Pi^2/2 - 235/16.

Sum_{n>=2} (-1)^n/a(n) = 3*Pi^2/4 - 117/16. (End)

E.g.f.: (1/36)*x^2*(180 + 300*x + 135*x^2 + 21*x^3 + x^4)*exp(x). - G. C. Greubel, Feb 21 2025