A006604 Generalized Fibonacci numbers. (Formerly M3469)

1, 1, 4, 13, 53, 228, 1037, 4885, 23640, 116793, 586633, 2986616, 15377097, 79927913, 418852716, 2210503285, 11738292397, 62673984492, 336260313765, 1811960161517, 9802082905840, 53213718977777, 289817858570513, 1583076422786096, 8670574105626961

Offset

0,3

References

D. G. Rogers, A Schroeder triangle: three combinatorial problems, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..24.

Plouffe, Simon, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

Formula

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

n*a(n) = (-1/2*n+3/2)*a(n-5)+(7/2*n-6)*a(n-4) +(13/2*n-9)*a(n-1) +(-7/2*n+15/2) *a(n-2) +(-3*n+3)*a(n-3). - Simon Plouffe, Master's Thesis, UQAM, 1992

a(n) = sum(k=1..n/2+1, (k*sum(j=0..n-2*k+2, (-1)^j*2^(n-2*k-j+2)*C(n-k+2,j) * C(2*n-3*k-j+3,n-k+1)))/((n-k+2))). - Vladimir Kruchinin, Oct 22 2011

Mathematica

CoefficientList[Series[(1+x-2 x^2-Sqrt[1-6 x+x^2])/(2 (2 x-x^2-x^3+x^4)), {x, 0, 60}], x]  (* Harvey P. Dale, Mar 23 2011 *)

Prog

(Maxima) a(n):=sum((k*sum((-1)^j*2^(n-2*k-j+2)*binomial(n-k+2, j)*binomial(2*n-3*k-j+3, n-k+1), j, 0, n-2*k+2))/((n-k+2)), k, 1, n/2+1); // Vladimir Kruchinin, Oct 22 2011

Crossrefs

Sequence in context: A149465 A149466 A369226 * A082570 A145208 A149467

Adjacent sequences: A006601 A006602 A006603 * A006605 A006606 A006607

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Mar 23 2011

Status

approved