A154964 a(n) = 3*a(n-1) + 6*a(n-2), n>2, a(0)=1, a(1)=1, a(2)=5.

1, 1, 5, 21, 93, 405, 1773, 7749, 33885, 148149, 647757, 2832165, 12383037, 54142101, 236724525, 1035026181, 4525425693, 19786434165, 86511856653, 378254174949, 1653833664765, 7231026043989, 31616080120557, 138234396625605, 604399670600157, 2642605391554101

Offset: 0

Author

Philippe Deléham, Jan 18 2009

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is the number of words of length n-1 over the alphabet {1,2,3,4,5} such that no two even numbers appear consecutively. - Armend Shabani, Mar 01 2017

Links

• Index entries for linear recurrences with constant coefficients, signature (3,6).

Programs

Mathematica

{1}~Join~LinearRecurrence[{3, 6}, {1, 5}, 25] (* or *)
CoefficientList[Series[(1 - 2 x - 4 x^2)/(1 - 3 x - 6 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Mar 02 2017 *)

PARI

Vec((1-2*x-4*x^2)/(1-3*x-6*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012

Formula

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

a(n+1) = Sum_{k=0..n} A154929(n,k)*2^(n-k).

G.f.: Q(0)/6 +2/3 , where Q(k) = 1 + 1/(1 - x*(6*k+3 + 6*x )/( x*(6*k+6 + 6*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013

a(n) = A083858(n+1)/3, n>=1. - R. J. Mathar, Feb 06 2020