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

0,2

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).

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

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A000129, A000045, A054446, A001477.

Sequence in context: A057153 A084787 A121190 * A061667 A234270 A258911

Adjacent sequences: A054444 A054445 A054446 * A054448 A054449 A054450

Keyword

easy,nonn

Author

Wolfdieter Lang, Apr 27 2000

Status

approved