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a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.
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%I #13 Jul 31 2015 12:24:32

%S 1,1,2,3,8,14,40,72,208,376,1088,1968,5696,10304,29824,53952,156160,

%T 282496,817664,1479168,4281344,7745024,22417408,40553472,117379072,

%U 212340736,614604800,1111830528,3218112512,5821620224,16850255872

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 6, 0, -4).

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

%F a(n + 4) = 6*a(n + 2) - 4*a(n) [From _Richard Choulet_, Dec 06 2008]

%F a(n) = ( - 1/20*5^(1/2) + 1/16*5^(1/2)*2^(1/2) - 1/16*2^(1/2) + 1/4)*(sqrt(3 + sqrt(5)))^n + (1/20*5^(1/2) + 1/16*5^(1/2)*2^(1/2) + 1/16*2^(1/2) + 1/4)*(sqrt(3 - sqrt(5)))^n + ( - 1/20*5^(1/2) - 1/16*5^(1/2)*2^(1/2) + 1/16*2^(1/2) + 1/4)*( - (sqrt(3 + sqrt(5))))^n + (1/20*5^(1/2) - 1/16*5^(1/2)*2^(1/2) - 1/16*2^(1/2) + 1/4)*( - (sqrt(3 - sqrt(5))))^n [From _Richard Choulet_, Dec 07 2008]

%t LinearRecurrence[{0,6,0,-4},{1,1,2,3},50] (* or *) CoefficientList[ Series[ (-3x^3-4x^2+x+1)/(4x^4-6x^2+1),{x,0,50}],x] (* _Harvey P. Dale_, May 02 2011 *)

%Y Cf. A080876, A080878, A080879, A080880, A080881, A080882.

%Y Cf. A154626, A098648 (bisections). [From _R. J. Mathar_, Oct 26 2009]

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 22 2003