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A008515 5-dimensional centered cube numbers.

%I A008515 #44 Sep 08 2022 08:44:36
%S A008515 1,33,275,1267,4149,10901,24583,49575,91817,159049,261051,409883,
%T A008515 620125,909117,1297199,1807951,2468433,3309425,4365667,5676099,
%U A008515 7284101,9237733,11589975,14398967,17728249,21647001,26230283,31559275,37721517,44811149,52929151,62183583,72689825
%N A008515 5-dimensional centered cube numbers.
%C A008515 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
%C A008515 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
%H A008515 Vincenzo Librandi, <a href="/A008515/b008515.txt">Table of n, a(n) for n = 0..10000</a>
%H A008515 Tito Piezas, <a href="https://mathoverflow.net/questions/265688/">About a Ramanujan-Sata formula of level 10, a recurrence, and zeta(5)?</a>, Mathoverflow question asked Mar 27 2017.
%H A008515 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A008515 a(n) = n^5 + (n+1)^5 = 2*n^5 +5*n^4 +10*n^3 +10*n^2 +5*n +1.
%F A008515 From _Bruno Berselli_, Aug 25 2011:  (Start)
%F A008515 G.f.: (1+x)*(1 +26*x +66*x^2 +26*x^3 +x^4)/(1-x)^6.
%F A008515 a(n) = -a(-n-1).
%F A008515 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)
%F A008515 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
%p A008515 seq(n^5+(n+1)^5, n=0..40);
%t A008515 Sum[(Range[40]+j-2)^5, {j,2}] (* _G. C. Greubel_, Nov 09 2019 *)
%o A008515 (PARI) a(n) = n^5+(n+1)^5;
%o A008515 (Magma) [n^5+(n+1)^5: n in [0..40]]; // _Bruno Berselli_, Aug 25 2011
%o A008515 (Sage) [n^5+(n+1)^5 for n in (0..40)] # _G. C. Greubel_, Nov 09 2019
%o A008515 (GAP) List([0..40], n-> n^5+(n+1)^5); # _G. C. Greubel_, Nov 09 2019
%Y A008515 Apart from the first term, a subsequence of A088703.
%K A008515 nonn,easy
%O A008515 0,2
%A A008515 _N. J. A. Sloane_