Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #36 Jan 18 2025 11:22:47
%S 1,793,66377,1911234,28504515,271739011,1874885963,10136389172,
%T 45311985069,173957200405,589679082421,1802148522758,5045944649967,
%U 13108508706879,31915866810295,73427944186856,160710828298553,336507487921137,677266380588289,1315464522556810
%N Sum of next n 6th powers.
%H T. D. Noe, <a href="/A075667/b075667.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
%F a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^6.
%F a(n) = (21n^13 + 231n^11 + 693n^9 + 549n^7 - 126n^5 - 56n^3 + 32n)/1344. - _Charles R Greathouse IV_, Sep 17 2009
%F G.f.: x*(x^12 +779*x^11 +55366*x^10 +1053755*x^9 +7499895*x^8 +23228658*x^7 +33620292*x^6 +23228658*x^5 +7499895*x^4 +1053755*x^3 +55366*x^2 +779*x +1)/(x-1)^14. - _Colin Barker_, Jul 22 2012
%e a(1) = 1^6 = 1; a(2) = 2^6 + 3^6 = 793; a(3) = 4^6 + 5^6 + 6^6 = 66377; a(4) = 7^6 + 8^6 + 9^6 + 10^6 = 1911234.
%t i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=6; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
%t With[{nn=20},Total/@TakeList[Range[(nn(nn+1))/2]^6,Range[nn]]] (* or *) LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{1,793,66377,1911234,28504515,271739011,1874885963,10136389172,45311985069,173957200405,589679082421,1802148522758,5045944649967,13108508706879},20] (* _Harvey P. Dale_, Mar 29 2022 *)
%Y Cf. A001014 (6th powers).
%Y Cf. A006003, A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).
%K nonn,easy,changed
%O 1,2
%A _Zak Seidov_, Sep 24 2002