OFFSET
0,2
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 219.
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).
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
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 *)
PROG
(SageMath) [i^4+(i+1)^4 for i in range(0, 36)] # Zerinvary Lajos, Jul 03 2008
(Magma) [(n+1)^4+n^4: n in [0..30]]; // Vincenzo Librandi, Aug 27 2011
(PARI) a(n) = n^4 + (n+1)^4; \\ Altug Alkan, Aug 01 2018
(GAP) List([0..30], n->n^4+(n+1)^4); # Muniru A Asiru, Aug 02 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved