A098588 a(n) = 2^n for n = 0..4; for n > 4, a(n) = 2*a(n-1) + a(n-5).

1, 2, 4, 8, 16, 33, 68, 140, 288, 592, 1217, 2502, 5144, 10576, 21744, 44705, 91912, 188968, 388512, 798768, 1642241, 3376394, 6941756, 14272024, 29342816, 60327873, 124032140, 255006036, 524284096, 1077911008, 2216149889

Offset

0,2

Comments

a(n) equals the number of n-length words on {0,1,2} such that 0 appears only in a run whose length is a multiple of 5. - Milan Janjic, Feb 17 2015

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,1).

Formula

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

a(n) = Sum_{k=0..floor(n/4)} Sum_{i=0..n} binomial(n-4k, i)binomial(i, k).

G.f.: G(0), where G(k)= 1 + x*(2+x^4)/(1 - x*(2+x^4)/(x*(2+x^4) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013

Lim_{n->infinity} a(n)/a(n+1) = 0.486389... is a real root of -1 + 2Z + Z^5 = 0. - Sergei N. Gladkovskii, Jul 03 2013

Mathematica

CoefficientList[Series[1/(1-2*x-x^5), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, 0, 0, 0, 1}, {1, 2, 4, 8, 16}, 50] (* G. C. Greubel, Feb 03 2018 *)

Prog

(PARI) x='x+O('x^30); Vec(1/(1-2*x-x^5)) \\ G. C. Greubel, Feb 03 2018
(Magma) I:=[1, 2, 4, 8, 16]; [n le 5 select I[n] else 2*Self(n-1) +Self(n-5): n in [1..30]]; // G. C. Greubel, Feb 03 2018

Crossrefs

Cf. A008998, A008999.

Sequence in context: A308808 A324406 A182442 * A367715 A126683 A005821

Adjacent sequences: A098585 A098586 A098587 * A098589 A098590 A098591

Keyword

nonn,easy

Author

Paul Barry, Sep 16 2004

Status

approved