A054447 Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).

1, 3, 9, 26, 73, 201, 545, 1460, 3873, 10191, 26633, 69198, 178889, 460437, 1180545, 3016552, 7684481, 19522203, 49473097, 125093506, 315654537, 795016545, 1998909985, 5017895196, 12578040097, 31485713511, 78716283081, 196563649718, 490301138569, 1221726409005

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000

Keywords

• easy
• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2604
• Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
• Index entries for linear recurrences with constant coefficients, signature (4, -2, -4, -1).

Crossrefs

Cf. A000129, A000045, A054446, A001477.

Programs

Mathematica

LinearRecurrence[{4, -2, -4, -1}, {1, 3, 9, 26}, 30] (* Michael De Vlieger, Jun 23 2020 *)

Maxima

a(n):=sum(k*sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n); /* Vladimir Kruchinin, Sep 06 2010 */

Formula

a(n) = Sum_{m=0..n} A054446(n,m) = ((n+1)*P(n+2)+(2-n)*P(n+1))/4, with P(n)=A000129(n) (Pell numbers).

G.f.: Pell(x)/(1-x*Fib(x)) = (Pell(x)^2)/Fib(x), with Pell(x)= 1/(1-2*x-x^2) = g.f. A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. A000045(n+1) (Fibonacci numbers without 0).

a(n) = Sum_(k*Sum_(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n), n>0. - Vladimir Kruchinin, Sep 06 2010

a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4), a(0)=1, a(1)=3, a(2)=9, a(3)=26. - Philippe Deléham, Jan 22 2014

G.f.: (1-x-x^2)/(1-2*x-x^2)^2 = g(f(x))/x, where g is g.f. of A001477 and f is g.f. of A000045. - Oboifeng Dira, Jun 21 2020