A080244 -id:A080244 - cp's OEIS frontend

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

N. J. A. Sloane, Simon Plouffe

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

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

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.

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

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

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

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 *)

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 */

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

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

More terms from Emeric Deutsch, Feb 28 2004

A080245 Inverse of coordination sequence array A113413.

Original entry on oeis.org

1, -2, 1, 6, -4, 1, -22, 16, -6, 1, 90, -68, 30, -8, 1, -394, 304, -146, 48, -10, 1, 1806, -1412, 714, -264, 70, -12, 1, -8558, 6752, -3534, 1408, -430, 96, -14, 1, 41586, -33028, 17718, -7432, 2490, -652, 126, -16, 1

Offset: 0

Paul Barry, Feb 13 2003

Formal inverse of A035607 when written as lower triangular matrix 1 2 1 2 4 1 ...

			Rows are {1}, {-2, 1}, {6, -4, 1}, {-22, 16, -6, 1}, ....
From _Paul Barry_, Apr 28 2009: (Start)
Triangle begins
  1,
  -2, 1,
  6, -4, 1,
  -22, 16, -6, 1,
  90, -68, 30, -8, 1,
  -394, 304, -146, 48, -10, 1,
  1806, -1412, 714, -264, 70, -12, 1
Production matrix is
  -2, 1,
  2, -2, 1,
  -2, 2, -2, 1,
  2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 1,
  2, -2, 2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 2, -2, 1 (End)

Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 7.

Row sums are signed little Schroeder numbers A080243. Diagonal sums are given by A080244.
Cf. A035607, A080243, A080244, A006603, A001003.
Cf. A000007 A084938.
Essentially same triangle as A033877 but with rows read in reversed order.

Essentially the same as the triangle T(n, k), for n>0 and k>0, given by [0, -2, -1, -2, -1, -2, -1, -2, ...] DELTA A000007. Triangle (unsigned) given by [0, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.

Riordan array ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2).

cp's OEIS Frontend

A006603 Generalized Fibonacci numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

SageMath

Formula

Extensions