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%I A075666 #26 Mar 23 2024 12:19:57
%S A075666 1,275,11925,208624,2078375,14118201,72758875,304553600,1084203549,
%T A075666 3390961375,9540835601,24582546000,58801331875,131987718149,
%U A075666 280410672375,567799960576,1102105900025,2060382328875,3724847929549,6534040766000,11154010982751,18575718271825
%N A075666 Sum of next n 5th powers.
%H A075666 T. D. Noe, <a href="/A075666/b075666.txt">Table of n, a(n) for n = 1..1000</a>
%H A075666 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A075666 a(1) = 1; a(n) = sum(i^s, {i, n(n-1)/2+1, n(n-1)/2+1+n}).
%F A075666 a(n) = (3n^11 + 25n^9 + 53n^7 + 23n^5 - 8n^3)/96.
%F A075666 G.f.: x*(x^10 +263*x^9 +8691*x^8 +83454*x^7 +301932*x^6 +458718*x^5 +301932*x^4 +83454*x^3 +8691*x^2 +263*x+1) / (x-1)^12. [_Colin Barker_, Jul 22 2012]
%e A075666 s=5; a(1) = 1^s = 1; a(2) = 2^s + 3^s = 2^5 + 3^5 = 275; a(3) = 4^s + 5^s + 6^s = 11925, a(4) = 7^s + 8^s + 9^s + 10^3 = 208624.
%t A075666 i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=5; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
%t A075666 nn=30;With[{p5=Range[((nn+1)(nn+2))/2]^5},Join[{1},Table[Total[Take[p5,{(n(n+1))/2+1,((n+1)(n+2))/2}]],{n,nn}]]] (* _Harvey P. Dale_, Mar 09 2014 *)
%t A075666 Module[{nn=25,p5},p5=Range[(nn(nn+1))/2]^5;Total/@TakeList[p5,Range[nn]]] (* _Harvey P. Dale_, Oct 13 2023 *)
%Y A075666 Cf. A072474 (s=2), A075664 - A075670 (s=3-10), A075671 (s=n).
%K A075666 nonn,easy
%O A075666 1,2
%A A075666 _Zak Seidov_, Sep 24 2002
%E A075666 Formula from _Charles R Greathouse IV_, Sep 17 2009

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