A006604 Generalized Fibonacci numbers.

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

Author

N. J. A. Sloane

Keywords

• nonn
• changed

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

Crossrefs

Cf. A006318, A006603.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( (1+x-2*x^2-Sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)) )); // G. C. Greubel, Aug 29 2025

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 *)

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 */

SageMath

def A006604_list(prec):
    P.= PowerSeriesRing(QQ, prec)
    return P( (1+x-2*x^2-sqrt(1-6*x+x^2))/(2*(2*x-x^2-x^3+x^4)) ).list()
print(A006604_list(41)) # G. C. Greubel, Aug 29 2025

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/(n-k+2))*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). - Vladimir Kruchinin, Oct 22 2011

G.f.: (1 - x + f(x))/(2 - x - x^2 + x^3), where f(x) is the gf of A006318. - G. C. Greubel, Aug 29 2025

Extensions

More terms from Harvey P. Dale, Mar 23 2011