A080877 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=1, a(2)=2.

1, 1, 2, 3, 8, 14, 40, 72, 208, 376, 1088, 1968, 5696, 10304, 29824, 53952, 156160, 282496, 817664, 1479168, 4281344, 7745024, 22417408, 40553472, 117379072, 212340736, 614604800, 1111830528, 3218112512, 5821620224, 16850255872

Offset

0,3

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -4).

Formula

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

a(n + 4) = 6*a(n + 2) - 4*a(n) - Richard Choulet, Dec 06 2008

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 - Richard Choulet, Dec 07 2008

Mathematica

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 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 0, 6, 0]^n*[1; 1; 2; 3])[1, 1] \\ Charles R Greathouse IV, May 16 2026

Crossrefs

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

Cf. A154626, A098648 (bisections).

Sequence in context: A129700 A197466 A049344 * A007165 A107321 A005316

Adjacent sequences: A080874 A080875 A080876 * A080878 A080879 A080880

Keyword

nonn,easy

Author

Paul D. Hanna, Feb 22 2003

Status

approved