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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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.

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

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 *)

PROG

(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 */

(Magma)

R<x>:=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

(SageMath)

def A006603_list(prec):

P.<x> = 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

CROSSREFS

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

Sequence in context: A150568 A102319 A367236 * A080244 A124542 A003447

Adjacent sequences: A006600 A006601 A006602 * A006604 A006605 A006606