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A385121 a(n+1) = 12*a(n) - a(n-1), a(0) = a(1) = 2, a(n) = a(1-n).

%I A385121 #30 Aug 29 2025 10:11:26
%S A385121 2,2,22,262,3122,37202,443302,5282422,62945762,750066722,8937854902,
%T A385121 106504192102,1269112450322,15122845211762,180205030090822,
%U A385121 2147337515878102,25587845160446402,304906804409478722,3633293807753298262,43294618888630100422
%N A385121 a(n+1) = 12*a(n) - a(n-1), a(0) = a(1) = 2, a(n) = a(1-n).
%C A385121 If x = 2, y = 6, z = a(n), w = a(n+1), then x^2+y^2+z^2+w^2 = x*y*z*w.
%H A385121 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-1).
%F A385121 G.f.: (2 - 22*x)/(1 - 12*x + x^2).
%F A385121 0 = 40 + a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 for all n in Z.
%F A385121 a(n) = 2 * A077417(n-1).
%F A385121 E.g.f.: 2*exp(6*x)*(7*cosh(sqrt(35)*x) - sqrt(35)*sinh(sqrt(35)*x))/7. - _Stefano Spezia_, Aug 29 2025
%e A385121 G.f. = 2 + 2*x + 22*x^2 + 262*x^3 + 3122*x^4 + 37202*x^5 + ...
%t A385121 a[ n_] := Which[n<1, a[1-n], n==1, 2, True, 12*a[n-1] - a[n-2]];
%o A385121 (PARI) {a(n) = if(n<1, a(1-n), n==1, 2, 12*a(n-1) - a(n-2))};
%Y A385121 Cf. A061292, A077417.
%K A385121 nonn,easy,changed
%O A385121 0,1
%A A385121 _Michael Somos_, Jun 18 2025