A071270 a(n) = n^2*(2*n^2 + 1)/3.

0, 1, 12, 57, 176, 425, 876, 1617, 2752, 4401, 6700, 9801, 13872, 19097, 25676, 33825, 43776, 55777, 70092, 87001, 106800, 129801, 156332, 186737, 221376, 260625, 304876, 354537, 410032, 471801, 540300, 616001, 699392, 790977, 891276, 1000825, 1120176

Offset: 0

Author

N. J. A. Sloane, Jun 12 2002

Keywords

• nonn
• easy

References

• T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A001105, A071238.

Programs

Magma

[n^2*(2*n^2+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Maple

A071270:=n->(n^2)*(2*n^2+1)/3; seq(A071270(n), n=0..100); # Wesley Ivan Hurt, Nov 14 2013

Mathematica

Table[(n^2)(2n^2+1)/3, {n,0,100}] (* Wesley Ivan Hurt, Nov 14 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,57,176},50] (* Harvey P. Dale, Jan 09 2016 *)

R

a <- c(0, 1, 12, 57, 176)
for(n in (length(a)+1):30)
     a[n] <- 5*a[n-1]-10*a[n-2]+10*a[n-3]-5*a[n-4]+a[n-5]
a # Yosu Yurramendi, Sep 03 2013

SageMath

def A071270(n): return binomial(2*n^2 + 1,2)/3
[A071270(n) for n in range(41)] # G. C. Greubel, Sep 13 2024

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=12, a(3)=57, a(4)=176. - Yosu Yurramendi, Sep 03 2013

a(n) = A000217(A001105(n))/ 3. - Michel Marcus, Mar 02 2018

From G. C. Greubel, Sep 13 2024: (Start)

G.f.: x*(1 + 7*x + 7*x^2 + x^3)/(1-x)^5.

E.g.f.: (1/3)*x*(3 + 15*x + 12*x^2 + 2*x^3)*exp(x). (End)