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%I #23 Mar 14 2023 12:57:32
%S 0,6,168,4374,113568,2948406,76545000,1987221606,51591216768,
%T 1339384414374,34772403556968,902743108066806,23436548406180000,
%U 608447515452613206,15796198853361763368,410092722671953234374,10646614590617422330368,276401886633381027355206
%N a(0)=0, a(1)=6 and a(n) = 26*a(n-1) - a(n-2) + 12 for n > 1.
%C Solutions to X*(X+1)=42*Y^2 with Y=A097309. - _R. J. Mathar_, Mar 14 2023
%H Seiichi Manyama, <a href="/A322708/b322708.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-27,1).
%F sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(7) + sqrt(6))^n.
%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(7) - sqrt(6))^n.
%F a(n) = 27*a(n-1) - 27*a(n-2) + a(n-3) for n > 2.
%F From _Colin Barker_, Dec 24 2018: (Start)
%F G.f.: 6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)).
%F a(n) = ((13+2*sqrt(42))^(-n) * (-1+(13+2*sqrt(42))^n)^2) / 4.
%F (End)
%F 2*a(n) = A097308(n)-1. - _R. J. Mathar_, Mar 14 2023
%e (sqrt(7) + sqrt(6))^2 = 13 + 2*sqrt(42) = sqrt(169) + sqrt(168). So a(2) = 168.
%t LinearRecurrence[{27,-27,1},{0,6,168},20] (* _Harvey P. Dale_, Apr 30 2022 *)
%o (PARI) concat(0, Vec(6*x*(1 + x) / ((1 - x)*(1 - 26*x + x^2)) + O(x^20))) \\ _Colin Barker_, Dec 24 2018
%Y Row 6 of A322699.
%K nonn,easy
%O 0,2
%A _Seiichi Manyama_, Dec 24 2018
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