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a(n) = 12*A002605(n).
0

%I #14 Apr 11 2024 18:18:17

%S 0,12,24,72,192,528,1440,3936,10752,29376,80256,219264,599040,1636608,

%T 4471296,12215808,33374208,91180032,249108480,680577024,1859371008,

%U 5079896064,13878534144,37916860416,103590789120,283015299072,773212176384,2112454950912

%N a(n) = 12*A002605(n).

%C The case k=2 in a family of sequences a(n)=G(k,n), G(k,0)=0, G(k,1)=k*(k+4), G(k,n)=k*G(k,n-1)+k*G(k,n-2).

%C The Binet formula is G(k,n) = (c^n-b^n)*d where d=sqrt(k*(k+4)); c=(k+d)/2; b=(k-d)/2.

%C The generating functions are k*(k+4)*x/(1-k*x-k*x^2).

%C The case k=1 is A022088.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).

%F Binet formula: a(n) = 2*2^n*((-1+3^(1/2))^(-n)-(-1)^n*(1+3^(1/2))^(-n))*3^(1/2) .

%F G.f.: 12*x/(1-2*x-2*x^2). a(n) = 2*a(n-1)+2*a(n-2).

%t LinearRecurrence[{2,2},{0,12},30] (* _Harvey P. Dale_, Mar 06 2023 *)

%Y Cf. A002605, A022088.

%K nonn,easy

%O 0,2

%A Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

%E Edited and extended by _R. J. Mathar_, Jan 23 2010