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 A102592 #21 Aug 23 2024 08:17:23
%S A102592 1,8,80,832,8704,91136,954368,9994240,104660992,1096024064,
%T A102592 11477712896,120196169728,1258710630400,13181388849152,
%U A102592 138037296103424,1445545331654656,15137947242201088,158526641599938560
%N A102592 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*5^(n-k).
%C A102592 In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).
%H A102592 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-16).
%F A102592 G.f.:(1-4x)/(1-12x+16x^2);
%F A102592 a(n) = 12*a(n-1) - 16*a(n-2);
%F A102592 a(n) = sqrt(5)*(sqrt(5)-1)^(2n+1)/10 + sqrt(5)*(sqrt(5)+1)^(2n+1)/10.
%F A102592 a(n) = Sum_{k=0..n} binomial(2n+1, k+1)*5^k. - _Paul Barry_, May 27 2005
%F A102592 a(n) = 4^(n+1)*A001519(n+1). - _N. J. A. Sloane_, Apr 13 2011
%F A102592 a(n) = 5^n* 2F1(-n-1/2, -n ; 1/2 ; 1/5). - _R. J. Mathar_, Aug 23 2024
%Y A102592 Cf. A066443, A102591.
%K A102592 easy,nonn
%O A102592 0,2
%A A102592 _Paul Barry_, Jan 22 2005