This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075665 #48 Jan 22 2025 06:04:56
%S A075665 1,97,2177,23058,152979,738835,2839571,9191876,26037717,66301333,
%T A075665 154762069,336050870,686502375,1331121351,2467171687,4396168328,
%U A075665 7566347369,12628007049,20504452585,32481640666,50320004987,76392352443,113852150523,166836980044,240712403645
%N A075665 Sum of next n 4th powers.
%H A075665 Kelvin Voskuijl, <a href="/A075665/b075665.txt">Table of n, a(n) for n = 1..10000</a>
%H A075665 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A075665 a(n) = Sum_{i=n*(n-1)/2+1..n*(n-1)/2+n} i^4.
%F A075665 a(n) = (15*n^9 + 90*n^7 + 123*n^5 + 20*n^3 - 8*n)/240. - _Charles R Greathouse IV_, Sep 17 2009
%F A075665 G.f.: x*(1+87*x+1252*x^2+5533*x^3+8934*x^4+5533*x^5+1252*x^6+87*x^7+x^8)/ (1-x)^10. - _Colin Barker_, May 25 2012
%e A075665 a(1) = 1^4 = 1; a(2) = 2^4 + 3^4 = 97; a(3) = 4^4 + 5^4 + 6^4 = 2177; a(4) = 7^4 + 8^4 + 9^4 + 10^4 = 23058.
%t A075665 i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=4; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
%t A075665 Table[Total[Range[(n(n+1))/2+1,((n+1)(n+2))/2]^4],{n,0,20}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,97,2177,23058,152979,738835,2839571,9191876,26037717,66301333},30] (* _Harvey P. Dale_, Dec 18 2015 *)
%Y A075665 Cf. A000583 (4th powers).
%Y A075665 Cf. A006003 (for natural numbers), A072474 (for squares), A075664 - A075671 (3rd to 10th powers), A069876 (n-th powers).
%K A075665 nonn,easy
%O A075665 1,2
%A A075665 _Zak Seidov_, Sep 24 2002