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%I #28 Mar 16 2023 08:17:10
%S 0,5,120,2645,58080,1275125,27994680,614607845,13493377920,
%T 296239706405,6503780163000,142786923879605,3134808545188320,
%U 68823001070263445,1510971215000607480,33172543728943101125,728284990821747617280,15989097254349504479045
%N a(0)=0, a(1)=5 and a(n) = 22*a(n-1) - a(n-2) + 10 for n > 1.
%C Solutions to X*(X+1)=30*Y^2 with Y=A077421. - _R. J. Mathar_, Mar 14 2023
%H Colin Barker, <a href="/A322707/b322707.txt">Table of n, a(n) for n = 0..700</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (23,-23,1).
%F sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(6) + sqrt(5))^n.
%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(6) - sqrt(5))^n.
%F a(n) = 23*a(n-1) - 23*a(n-2) + a(n-3) for n > 2.
%F From _Colin Barker_, Dec 24 2018: (Start)
%F G.f.: 5*x*(1 + x) / ((1 - x)*(1 - 22*x + x^2)).
%F a(n) = ((11+2*sqrt(30))^(-n) * (-1+(11+2*sqrt(30))^n)^2) / 4.
%F (End)
%F 2*a(n) = A077422(n)-1. - _R. J. Mathar_, Mar 16 2023
%e (sqrt(6) + sqrt(5))^2 = 11 + 2*sqrt(30) = sqrt(121) + sqrt(120). So a(2) = 120.
%o (PARI) concat(0, Vec(5*x*(1 + x) / ((1 - x)*(1 - 22*x + x^2)) + O(x^20))) \\ _Colin Barker_, Dec 24 2018
%Y Row 5 of A322699.
%Y Cf. A188930 (sqrt(5)+sqrt(6)).
%K nonn,easy
%O 0,2
%A _Seiichi Manyama_, Dec 24 2018
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