A008514 4-dimensional centered cube numbers.

1, 17, 97, 337, 881, 1921, 3697, 6497, 10657, 16561, 24641, 35377, 49297, 66977, 89041, 116161, 149057, 188497, 235297, 290321, 354481, 428737, 514097, 611617, 722401, 847601, 988417, 1146097, 1321937, 1517281, 1733521, 1972097, 2234497, 2522257, 2836961, 3180241

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Summation of n^4 taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 27 2009

The primes in this sequence are given by A152913. - Jonathan Vos Post, Aug 17 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A002061, A047838, A152913.

Programs

GAP

List([0..30],n->n^4+(n+1)^4); # Muniru A Asiru, Aug 02 2018

Magma

[(n+1)^4+n^4: n in [0..30]]; // Vincenzo Librandi, Aug 27 2011

Maple

seq(n^4+(n+1)^4, n=0..40);

Mathematica

Total/@Partition[Range[0, 30]^4, 2, 1] (* or *) LinearRecurrence[{5,-10, 10,-5,1}, {1,17,97,337,881}, 30] (* Harvey P. Dale, Jan 28 2013 *)

PARI

a(n) = n^4 + (n+1)^4; \\ Altug Alkan, Aug 01 2018

Sage

[i^4+(i+1)^4 for i in range(0,36)] # Zerinvary Lajos, Jul 03 2008

Formula

a(n) = n^4 + (n+1)^4.

a(n) = 2*n^4 + 4*n^3 + 6*n^2 + 4*n + 1. - Al Hakanson (hawkuu(AT)gmail.com), May 27 2009, corrected R. J. Mathar, May 29 2009

G.f.: (1+10*x+x^2)*(1+x)^2/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 09 2009

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0) = 1, a(1) = 17, a(2) = 97, a(3) = 337, a(4) = 881. - Harvey P. Dale, Jan 28 2013

a(n) = 4*(n+n^2) + 2*(n+n^2)^2 + 1. - Avi Friedlich, Mar 31 2015

a(n) = 2*A002061(n+1)^2 - 1. - Bruce J. Nicholson, Apr 14 2017

a(n) = A047838(2*(n^2+n+1)). - David James Sycamore, Aug 01 2018

E.g.f.: (1 + 16*x + 32*x^2 + 16*x^3 + 2*x^4)*exp(x). - G. C. Greubel, Nov 09 2019

Sum_{n>=0} 1/a(n) = (tanh((sqrt(2)-1)*Pi/2)*Pi*(2+sqrt(2)) - tanh((sqrt(2)+1)*Pi/2)*Pi*(2-sqrt(2)))/4. - Amiram Eldar, Sep 20 2022