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 A007750 #37 Dec 29 2024 16:10:42
%S A007750 0,1,7,24,120,391,1921,6240,30624,99457,488071,1585080,7778520,
%T A007750 25261831,123968257,402604224,1975713600,6416405761,31487449351,
%U A007750 102259887960,501823476024,1629741801607,7997688167041,25973608937760
%N A007750 Nonnegative integers n such that n^2*(n+1)*(2*n+1)^2*(7*n+1)/36 is a square.
%C A007750 n^2*(n+1)*(2*n+1)^2*(7*n+1)/36 = Sum(i=1..n, i^2) * Sum(i=n+1..2*n, i^2) = A000330(n)*(A000330(2*n)-A000330(n)) = A000330(n)*n*(2*n+1)*(7*n+1)/6. - _Michael Somos_, Jul 27 2002
%H A007750 G. C. Greubel, <a href="/A007750/b007750.txt">Table of n, a(n) for n = 0..1000</a>
%H A007750 K. R. S. Sastry, <a href="https://www.jstor.org/stable/2687624">Problem 533</a> The College Mathematics Journal, 25, issue 4, 1994, p. 334.
%H A007750 K. R. S. Sastry, <a href="https://www.jstor.org/stable/2687039">Square Products of Sums of Squares</a> The College Mathematics Journal, 26, issue 4, 1995, p. 333.
%H A007750 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,16,-16,-1,1).
%F A007750 From _Michael Somos_, Jul 27 2002: (Start)
%F A007750 G.f.: x * (1 + 6*x + x^2) / ((1 - x) * (1 - 16*x^2 + x^4)).
%F A007750 a(n) = 16 * a(n-2) - a(n-4) + 8. (End)
%F A007750 From _G. C. Greubel_, Feb 10 2020: (Start)
%F A007750 a(2*n) = (4*ChebyshevU(n,8) - 11*ChebyshevU(n-1,8) - 4)/7 = A007751(n).
%F A007750 a(2*n+1) = (11*ChebyshevU(n,8) - 4*ChebyshevU(n-1,8) - 4)/7 = A007752(n+1). (End)
%p A007750 m:=30; S:=series(x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 10 2020
%t A007750 CoefficientList[Series[x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jan 15 2017 *)
%t A007750 Table[If[EvenQ[n], (4*ChebyshevU[n/2,8] -11*ChebyshevU[(n-2)/2,8] -4)/7, (11*ChebyshevU[(n-1)/2,8] -4*ChebyshevU[(n-3)/2,8] -4)/7], {n,0,30}] (* _G. C. Greubel_, Feb 10 2020 *)
%o A007750 (PARI) {a(n) = if( n<0, a(-1-n), if( n<2, n>0, 16 * a(n-2) - a(n-4) + 8))} /* _Michael Somos_, Jul 27 2002 */
%o A007750 (PARI) {a(n) = local(w); if( n<0, 0, w = 8 + 3*quadgen(28); n = ((n+1)\2) * (-1)^(n%2); imag(w^n) + 4 * (real(w^n) - 1) / 7)} /* _Michael Somos_, Jul 27 2002 */
%o A007750 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)) )); // _G. C. Greubel_, Feb 10 2020
%o A007750 (Sage)
%o A007750 def A007750_list(prec):
%o A007750 P.<x> = PowerSeriesRing(ZZ, prec)
%o A007750 return P( x*(1+6*x+x^2)/((1-x)*(1-16*x^2+x^4)) ).list()
%o A007750 A007750_list(30) # _G. C. Greubel_, Feb 10 2020
%o A007750 (GAP) a:=[0,1,7,24,120];; for n in [6..30] do a[n]:=a[n-1]+16*a[n-2]-16*a[n-3] -a[n-4]+a[n-5]; od; a; # _G. C. Greubel_, Feb 10 2020
%Y A007750 Cf. A007751, A007752, A077412.
%K A007750 nonn,easy
%O A007750 0,3
%A A007750 John C. Hallyburton, Jr. (hallyb(AT)vmsdev.enet.dec.com)
%E A007750 Edited by _Michael Somos_, Jul 27 2002