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%I #18 Jan 25 2023 12:51:59
%S 1,8,42,204,972,4608,21816,103248,488592,2312064,10940832,51772608,
%T 244990656,1159308288,5485905792,25959585024,122842075392,
%U 581294942208,2750717200896,13016533552128,61594898107392,291470187331584,1379251735345152,6526689288081408,30884625316417536
%N Expansion of (1 + 2*x)/(1 - 6*x + 6*x^2).
%C Satisfies recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=1, b(0)=2.
%C More generally, for the recurrence relations system a(n) = 4*a(n-1) + 2*b(n-1), b(n) = 2*b(n-1) + a(n-1), a(0)=k, b(0)=m solution is a(n) = (((sqrt(3) - 1)*k - 2*m)*(3 - sqrt(3))^n + (sqrt(3)*k + k + 2*m)*(3 + sqrt(3))^n)/(2*sqrt(3)), b(n) = ((-k + sqrt(3)*m + m)*(3 - sqrt(3))^n + (k + (sqrt(3) - 1)*m)*(3 + sqrt(3))^n)/(2*sqrt(3)).
%C Convolution of A030192 and {1, 2, 0, 0, 0, 0, 0, ...}.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6)
%F O.g.f.: (1 + 2*x)/(1 - 6*x + 6*x^2).
%F E.g.f.: (5*sqrt(3)*sinh(sqrt(3)*x) + 3*cosh(sqrt(3)*x))*exp(3*x)/3.
%F a(n) = 6*a(n-1) - 6*a(n-2).
%F a(n) = ((-5 + sqrt(3))*(3 - sqrt(3))^n + (5 + sqrt(3))*(3 + sqrt(3))^n)/(2*sqrt(3)).
%F Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(3) = A165663.
%F a(n) = A030192(n)+2*A030192(n-1). - _R. J. Mathar_, Jan 25 2023
%p a:=series((1+2*x)/(1-6*x+6*x^2),x=0,25): seq(coeff(a,x,n),n=0..24); # _Paolo P. Lava_, Mar 27 2019
%t LinearRecurrence[{6, -6}, {1, 8}, 25]
%t CoefficientList[Series[(1 + 2 x)/(1 - 6 x + 6 x^2), {x, 0, 24}], x] (* _Michael De Vlieger_, Aug 26 2016 *)
%o (PARI) Vec((1+2*x)/(1-6*x+6*x^2) + O(x^99)) \\ _Altug Alkan_, Aug 26 2016
%Y Cf. A030192, A165663.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Aug 26 2016
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