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%I #13 Mar 04 2024 01:10:51
%S 3,8,30,132,612,2880,13608,64368,304560,1441152,6819552,32270400,
%T 152705088,722608128,3419418240,16180860672,76568654592,362326763520,
%U 1714548653568,8113331340288,38392696120320,181676188680192
%N a(n) = (4+sqrt(3))*(3+sqrt(3))^n + (4-sqrt(3))*(3-sqrt(3))^n.
%C Equals b(k) = ((9+sqrt(3))*(3-sqrt(3))^k+(9-sqrt(3))*(3+sqrt(3))^k)/6 for k >= 0. - _Klaus Brockhaus_, Jul 14 2009
%C Third binomial transform of A162852.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F a(n) = 6*a(n-1) - 6*a(n-2) for n > 0; a(-1) = 3, a(0) = 8.
%F G.f.: (3-10*x)/(x*(1-6*x+6*x^2)).
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n): n in [-1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 14 2009
%Y Cf. A162852 (3, -1, 9, -3, 27, -9, 81, ...).
%K nonn,easy
%O -1,1
%A Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009
%E Edited and extended beyond a(5)=13608 by _Klaus Brockhaus_, Jul 18 2009