OFFSET
0,3
COMMENTS
Partial sums of A006130 (with leading zero).
Specify a triangle by T(n,0) = T(n+1,1) = A001045(n) and T(n,k) = T(n-1,k-1) + T(n-1,k-2) + T(n-2,k-2) otherwise. Then T(n,n)= a(n-1). - J. M. Bergot, May 24 2013
LINKS
FORMULA
G.f.: x/((1-x)*(1-x-3*x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*3^k.
a(n) = (1/2 + sqrt(13)/2)^n*(1/6 + 7*sqrt(13)/78) + (1/6 - 7*sqrt(13)/78)*(1/2 - sqrt(13)/2)^n - 1/3.
a(n+1) = Sum_{k=0..n} C(k+1,n-k+1)*3^(n-k). - Paul Barry, May 21 2006
a(n) = a(n-1) + 3*a(n-2) + 1, n > 1. - Gary Detlefs, Jun 21 2010
G.f.: Q(0)*x/(2-2*x), where Q(k) = 1 + 1/(1 - x*(4*k+1 + 3*x)/( x*(4*k+3 + 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
MATHEMATICA
LinearRecurrence[{2, 2, -3}, {0, 1, 2}, 40] (* Harvey P. Dale, Mar 02 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 08 2004
STATUS
approved