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

2, 2, 22, 262, 3122, 37202, 443302, 5282422, 62945762, 750066722, 8937854902, 106504192102, 1269112450322, 15122845211762, 180205030090822, 2147337515878102, 25587845160446402, 304906804409478722, 3633293807753298262, 43294618888630100422

Offset: 0

Author

Michael Somos, Jun 18 2025

Keywords

• nonn
• easy
• changed

Comments

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.

Examples

			G.f. = 2 + 2*x + 22*x^2 + 262*x^3 + 3122*x^4 + 37202*x^5 + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (12,-1).

Crossrefs

Cf. A061292, A077417.

Programs

Mathematica

a[ n_] := Which[n<1, a[1-n], n==1, 2, True, 12*a[n-1] - a[n-2]];

PARI

{a(n) = if(n<1, a(1-n), n==1, 2, 12*a(n-1) - a(n-2))};

Formula

G.f.: (2 - 22*x)/(1 - 12*x + x^2).

0 = 40 + a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 for all n in Z.

a(n) = 2 * A077417(n-1).

E.g.f.: 2*exp(6*x)*(7*cosh(sqrt(35)*x) - sqrt(35)*sinh(sqrt(35)*x))/7. - Stefano Spezia, Aug 29 2025