A006603 Generalized Fibonacci numbers.

1, 2, 7, 26, 107, 468, 2141, 10124, 49101, 242934, 1221427, 6222838, 32056215, 166690696, 873798681, 4612654808, 24499322137, 130830894666, 702037771647, 3783431872018, 20469182526595, 111133368084892, 605312629105205, 3306633429423460, 18111655081108453

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

The Kn21 sums, see A180662, of the Schroeder triangle A033877 equal A006603(n) while the Kn3 sums equal A006603(2*n). The Kn22 sums, see A227504, and the Kn23 sums, see A227505, are also related to the sequence given above. - Johannes W. Meijer, Jul 15 2013

Typo on the right-hand side of Rogers's equation (1-x+x^2+x^3)*R^*(x) = R(x) + x: the sign in front of the x should be switched. - R. J. Mathar, Nov 23 2018

References

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

Crossrefs

Cf. A006318, A006603, A033877, A080244, A180662, A227504, A227505.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!( (1-x-2*x^2 -Sqrt(1-6*x+x^2))/(2*x*(1-x+x^2+x^3)) )); // G. C. Greubel, Oct 27 2024

Maple

A006603 := n-> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): seq(A006603(n), n=0..24); # Johannes W. Meijer, Jul 15 2013

Mathematica

CoefficientList[Series[(1-x-2x^2-Sqrt[1-6x+x^2])/(2x(1-x+x^2+x^3)),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

Maxima

a(n):=sum((k*sum(binomial(n-k+2,i)*binomial(2*n-3*k-i+3,n-k+1),i,0,n-2*k+2))/(n-k+2),k,1,n/2+1); /* Vladimir Kruchinin, Oct 23 2011 */

SageMath

def A006603_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x-2*x^2 -sqrt(1-6*x+x^2))/(2*x*(1-x+x^2+x^3)) ).list()
A006603_list(50) # G. C. Greubel, Oct 27 2024

Formula

a(n) = abs(A080244(n-1)).

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

G.f.: (A006318(x) - x)/(1 - x + x^2 + x^3).

a(n) = Sum_{k=1..floor(n/2)+1} k*(1/(n-k+2))*Sum_{i=0..n-2*k+2} C(n-k+2,i)*C(2*n-3*k-i+3,n-k+1). - Vladimir Kruchinin, Oct 23 2011

(n+1)*a(n) -(7*n-2)*a(n-1) +4*(2*n-1)*a(n-2) -6*(n-1)*a(n-3) -(5*n-1)*a(n-4) +(n-2)*a(n-5) = 0. - R. J. Mathar, Nov 23 2018

Extensions

More terms from Emeric Deutsch, Feb 28 2004