A122994 a(n) = a(n-1)+9*a(n-2) initialized with a(0)=1, a(1)=3.

1, 3, 12, 39, 147, 498, 1821, 6303, 22692, 79419, 283647, 998418, 3551241, 12537003, 44498172, 157331199, 557814747, 1973795538, 6994128261, 24758288103, 87705442452, 310530035379, 1099879017447, 3894649335858, 13793560492881, 48845404515603, 172987448951532

Offset: 0

Author

Roger L. Bagula, Sep 22 2006

Keywords

• nonn
• easy

Comments

The two roots of the denominator of the g.f. (for Binet's formula) are -0.393486... and 0.2823756...

Pisano period lengths: 1, 3, 1, 6, 6, 3, 6, 12, 1, 6, 10, 6, 84, 6, 6, 24,144, 3, 72, 6,... - R. J. Mathar, Aug 10 2012

Links

• Index entries for linear recurrences with constant coefficients, signature (1,9).

Crossrefs

Cf. A026597.

Programs

Mathematica

CoefficientList[Series[(-2 z - 1)/(9 z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

Formula

G.f.: -(1+2*x)/(-1+x+9*x^2). a(n) = A015445(n)+2*A015445(n-1). [R. J. Mathar, Aug 12 2009]

a(n) = (1/2+5*sqrt(37)/74) *(1/2+sqrt(37)/2)^(n-1) +(1/2-5*sqrt(37)/74) *(1/2-sqrt(37)/2)^(n-1). [Antonio Alberto Olivares, Jun 07 2011]

a(n) = Sum_{k, 0<=k<=n} A103631(n,k)*3^k. - Philippe Deléham, Dec 17 2011

a(n) = A015445(n) + 2*A015445(n-1), n>0. - Ralf Stephan, Jul 21 2013

Extensions

Definition replaced with the Deleham recurrence of Mar 2009 by the Assoc. Editors of the OEIS, Mar 12 2010