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

%I M2976 #42 Aug 23 2022 12:22:00

%S 1,1,1,3,14,85,626,5387,52882,582149,7094234,94730611,1374650042,

%T 21529197077,361809517954,6492232196699,123852300381986,

%U 2502521367966277,53379537613065002,1198434678728086019,28245547605034208074,697186985180529270101

%N Number of n-term 2-sided generalized Fibonacci sequences.

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

%H N. J. A. Sloane, <a href="/A005189/b005189.txt">Table of n, a(n) for n = 0..250</a>

%H C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, <a href="http://lipn.univ-paris13.fr/~banderier/Papers/mis-n-to-the-logn.pdf">Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics</a>, SIAM J. Discrete Math., 28(1), 342-371. (30 pages), DOI:10.1137/130916357. - From _N. J. A. Sloane_, Dec 23 2012

%H Peter C. Fishburn, Peter C. Marcus-Roberts, Fred S. Roberts, <a href="http://dx.doi.org/10.1137/0401034">Unique finite difference measurement</a>, SIAM J. Discrete Math. 1 (1988), no. 3, 334-354.

%H P. C. Fishburn, A. M. Odlyzko and F. S. Roberts, <a href="http://www.fq.math.ca/Scanned/27-4/fishburn.pdf">2-sided generalized Fibonacci sequences</a>, Fib. Quart., 27 (1989), 352-361.

%H Rui-Li Liu, Feng-Zhen Zhao, <a href="https://www.emis.de/journals/JIS/VOL21/Liu/liu19.html">New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.

%H Poloni, Federico; Del Corso, Gianna M. <a href="https://doi.org/10.1016/j.laa.2017.06.042">Counting Fiedler pencils with repetitions</a>. Linear Algebra Appl. 532, 463-499 (2017).

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

%F 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.

%F 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

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

%p A005189 :=proc(n) option remember;

%p if n <= 2 then 1 else 2*(n-1)*procname(n-1)-(n-2)^2*procname(n-2); fi; end;

%p [seq(A005189(n),n=0..20)]; # _N. J. A. Sloane_, Jul 10 2015

%t $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 *)

%o (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 */

%K nonn

%O 0,4

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Vladeta Jovovic_, Sep 05 2005