A005189 Number of n-term 2-sided generalized Fibonacci sequences. (Formerly M2976)

1, 1, 1, 3, 14, 85, 626, 5387, 52882, 582149, 7094234, 94730611, 1374650042, 21529197077, 361809517954, 6492232196699, 123852300381986, 2502521367966277, 53379537613065002, 1198434678728086019, 28245547605034208074, 697186985180529270101

Offset

0,4

References

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..250

C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, SIAM J. Discrete Math., 28(1), 342-371. (30 pages), DOI:10.1137/130916357. - From N. J. A. Sloane, Dec 23 2012

Peter C. Fishburn, Peter C. Marcus-Roberts, Fred S. Roberts, Unique finite difference measurement, SIAM J. Discrete Math. 1 (1988), no. 3, 334-354.

P. C. Fishburn, A. M. Odlyzko and F. S. Roberts, 2-sided generalized Fibonacci sequences, Fib. Quart., 27 (1989), 352-361.

Rui-Li Liu, Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.

Poloni, Federico; Del Corso, Gianna M.  Counting Fiedler pencils with repetitions. Linear Algebra Appl. 532, 463-499 (2017).

Formula

If n <= 2 then a(n) = 1 otherwise a(n) = 2*(n-1)*a(n-1)-(n-2)^2*a(n-2).

E.g.f.: (e*Ei(1/(x-1)) - e*Ei(-1)-1)/(e^(x/(x-1))*(x-1)), where Ei is the exponential integral function. - Jean-François Alcover, Sep 05 2015, after Fishburn et al.

0 = a(n)*(-24*a(n+2) + 99*a(n+3) - 78*a(n+4) + 17*a(n+5) - a(n+6)) + a(n+1)*(-15*a(n+2) + 84*a(n+3) - 51*a(n+4) + 6*a(n+5)) + a(n+2)*(-6*a(n+2) + 34*a(n+3) - 15*a(n+4)) + a(n+3)*(+10*a(n+3)) for all n in Z. - Michael Somos, Dec 02 2016

Example

G.f. = 1 + x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 626*x^6 + 5387*x^7 + ...

Maple

A005189 :=proc(n) option remember;
if n <= 2 then 1 else 2*(n-1)*procname(n-1)-(n-2)^2*procname(n-2); fi; end;
[seq(A005189(n), n=0..20)]; # N. J. A. Sloane, Jul 10 2015

Mathematica

$Assumptions = Element[x, Reals]; F[x_] := (E*ExpIntegralEi[1/(x-1)] - E*ExpIntegralEi[-1]-1)/(E^(x/(x-1))*(x-1)); Join[{1}, CoefficientList[ Normal[Series[F[x], {x, 0, 18}]], x]*Range[0, 18]!] (* Jean-François Alcover, Sep 05 2015 *)

Prog

(PARI) {a(n) = if(n<3, n>=0, 2*(n-1)*a(n-1) - (n-2)^2*a(n-2))}; /* Michael Somos, Dec 02 2016 */

Crossrefs

Sequence in context: A263187 A213628 A088716 * A331608 A331615 A317060

Adjacent sequences: A005186 A005187 A005188 * A005190 A005191 A005192

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Vladeta Jovovic, Sep 05 2005

Status

approved