A080880 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=2, a(2)=2.

1, 2, 2, 5, 6, 17, 22, 65, 86, 257, 342, 1025, 1366, 4097, 5462, 16385, 21846, 65537, 87382, 262145, 349526, 1048577, 1398102, 4194305, 5592406, 16777217, 22369622, 67108865, 89478486, 268435457, 357913942, 1073741825, 1431655766, 4294967297

Offset

0,2

Links

Table of n, a(n) for n=0..33.

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -4).

Formula

a(2n)=(4^n+2)/3, a(2n+1)=4^n+1.

G.f.: (-5*x^3 - 3*x^2 + 2*x + 1)/(4*x^4 - 5*x^2 + 1)

a(n) = 5/6 + 5/12*2^n - 1/6*( - 1)^n - 1/12*( - 2)^n - Richard Choulet, Dec 07 2008

a(n + 4) = 5*a(n + 2) - 4*a(n) - Richard Choulet, Dec 06 2008

Mathematica

LinearRecurrence[{0, 5, 0, -4}, {1, 2, 2, 5}, 40] (* Harvey P. Dale, Nov 30 2012 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -4, 0, 5, 0]^n*[1; 2; 2; 5])[1, 1] \\ Charles R Greathouse IV, May 16 2026
(PARI) Vec((-5*x^3-3*x^2+2*x+1)/(4*x^4-5*x^2+1)+O(x^99)) \\ Charles R Greathouse IV, May 16 2026

Crossrefs

Cf. A080876, A080877, A080878, A080879, A080881, A080882.

Sequence in context: A320296 A329908 A295198 * A305606 A369634 A120843

Adjacent sequences: A080877 A080878 A080879 * A080881 A080882 A080883

Keyword

nonn,easy

Author

Paul D. Hanna, Feb 22 2003

Extensions

More terms from Ralf Stephan, Jul 25 2003

Status

approved