A008515 5-dimensional centered cube numbers.

1, 33, 275, 1267, 4149, 10901, 24583, 49575, 91817, 159049, 261051, 409883, 620125, 909117, 1297199, 1807951, 2468433, 3309425, 4365667, 5676099, 7284101, 9237733, 11589975, 14398967, 17728249, 21647001, 26230283, 31559275, 37721517, 44811149, 52929151, 62183583, 72689825

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

These are never prime, as a(n) = (2n+1)*(n^4+2*n^3+4*n^2+3*n+1). - Jonathan Vos Post, Aug 18 2011

zeta(5) = 1 / (a(1) - 1^10 / (a(2) - 2^10 / (a(3) - 3^10 / ... ))) [From Tito Piezas III mathoverflow question 265688 comment]. - Michael Somos, Jul 06 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Tito Piezas, About a Ramanujan-Sata formula of level 10, a recurrence, and zeta(5)?, Mathoverflow question asked Mar 27 2017.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Apart from the first term, a subsequence of A088703.

Programs

GAP

List([0..40], n-> n^5+(n+1)^5); # G. C. Greubel, Nov 09 2019

Magma

[n^5+(n+1)^5: n in [0..40]]; // Bruno Berselli, Aug 25 2011

Maple

seq(n^5+(n+1)^5, n=0..40);

Mathematica

Sum[(Range[40]+j-2)^5, {j,2}] (* G. C. Greubel, Nov 09 2019 *)

PARI

a(n) = n^5+(n+1)^5;

Sage

[n^5+(n+1)^5 for n in (0..40)] # G. C. Greubel, Nov 09 2019

Formula

a(n) = n^5 + (n+1)^5 = 2*n^5 +5*n^4 +10*n^3 +10*n^2 +5*n +1.

From Bruno Berselli, Aug 25 2011: (Start)

G.f.: (1+x)*(1 +26*x +66*x^2 +26*x^3 +x^4)/(1-x)^6.

a(n) = -a(-n-1).

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). (End)

E.g.f.: (1 +32*x +105*x^2 +90*x^3 +25*x^4 +2*x^5)*exp(x). - G. C. Greubel, Nov 09 2019