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a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.
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%I #19 Jun 11 2024 10:57:07

%S 1,1,1,2,4,12,28,88,208,656,1552,4896,11584,36544,86464,272768,645376,

%T 2035968,4817152,15196672,35955712,113429504,268377088,846649344,

%U 2003193856,6319476736,14952042496,47169216512,111603564544

%N a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0) = 1, a(1) = 1, and a(2) = 1.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 8, 0, -4).

%F G.f.: (-6*x^3 - 7*x^2 + x + 1)/(4*x^4 - 8*x^2 + 1)

%F a(n + 4) = 8*a(n + 2) - 4*a(n). - _Richard Choulet_, Dec 06 2008

%F a(n) = (1/24 * 3^(1/2)) * (1 + sqrt(3))^n - (1/24 * 3^(1/2)) * (1 - sqrt(3))^n + (1/2 - 7/24 * 3^(1/2)) * (-(1 + sqrt(3)))^n + (1/2 + 7/24 * 3^(1/2))*(-(1 - sqrt(3)))^n. - _Richard Choulet_, Dec 06 2008

%t CoefficientList[Series[(-6x^3-7x^2+x+1)/(4x^4-8x^2+1),{x,0,40}],x] (* _Harvey P. Dale_, Mar 04 2011 *)

%Y Cf. A080871, A080877, A080878, A080879, A080880, A080881, A080882.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Feb 22 2003