A071239 a(n) = n*(n+1)*(n^2 + 2)/6.

0, 1, 6, 22, 60, 135, 266, 476, 792, 1245, 1870, 2706, 3796, 5187, 6930, 9080, 11696, 14841, 18582, 22990, 28140, 34111, 40986, 48852, 57800, 67925, 79326, 92106, 106372, 122235, 139810, 159216, 180576, 204017, 229670, 257670, 288156, 321271, 357162, 395980

Offset: 0

Author

N. J. A. Sloane, Jun 12 2002

Keywords

• nonn
• easy

Comments

Number of binary pattern classes with 4 black beads in the (2,n)-rectangular grid; two patterns are in the same class if one of them can be obtained by reflection or rotation of the other one. - Yosu Yurramendi, Sep 12 2008

This sequence is the case k=n+3 of b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6, which is the n-th k-gonal pyramidal number. Therefore, apart from 0, this sequence is the 3rd diagonal of the array in A080851. - Luciano Ancora, Apr 10 2015

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. A080851, A226048.

Programs

Magma

[n*(n+1)*(n^2+2)/6: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Mathematica

Table[(n(n+1)(n^2+2))/6,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,6,22,60},40] (* Harvey P. Dale, May 01 2013 *)

PARI

a(n)=n*(n+1)*(n^2+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

R

a <- vector()
    for(n in 1:40) a[n] <- (1/4)*(choose(2*n,4) + 3*choose(n,2))
    a
# Yosu Yurramendi and María Merino, Aug 21 2013

SageMath

def A071239(n): return binomial(n+1,2)*(n^2+2)//3
[A071239(n) for n in range(41)] # G. C. Greubel, Aug 06 2024

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n>4. - Harvey P. Dale, May 01 2013

a(n) = (binomial(2*n+2,4) + 3*binomial(n+1,2))/4 = (A053134(n-1) + 3*A000217(n))/4 . - Yosu Yurramendi and María Merino, Aug 21 2013

G.f.: x*(1+x+2*x^2) / (1-x)^5 . - R. J. Mathar, Aug 21 2013

E.g.f.: (1/6)*x*(6 + 12*x + 7*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 06 2024