A000128 A nonlinear binomial sum. (Formerly M1120 N0428)

1, 2, 4, 8, 16, 31, 58, 105, 185, 319, 541, 906, 1503, 2476, 4058, 6626, 10790, 17537, 28464, 46155, 74791, 121137, 196139, 317508, 513901, 831686, 1345888, 2177900, 3524140, 5702419, 9226966, 14929821, 24157253, 39087571, 63245353, 102333486

Offset

1,2

References

Ralph P. Grimaldi, A generalization of the Fibonacci sequence. Proceedings of the seventeenth Southeastern international conference on combinatorics, graph theory, and computing (Boca Raton, Fla., 1986). Congr. Numer. 54 (1986), 123-128. MR0885268 (89f:11030). - N. J. A. Sloane, Apr 08 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

T. D. Noe, Table of n, a(n) for n=1..201

D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 292-298.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

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

a(n) = F(n+4) - n*(n+1)/2 - 3, with F(n) = A000045(n). - Ralf Stephan, Aug 19 2004

a(n) = 2*a(n-1) - a(n-3) + n - 3, n > 3, and a(1) = 1, a(2) = 2, a(3) = 4. - Chunqing Liu, Sep 23 2023

Maple

A000128:=(1-2*z+z**2+z**3)/(z**2+z-1)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{4, -5, 1, 2, -1}, {1, 2, 4, 8, 16}, 40] (* Jean-François Alcover, Feb 04 2016 *)

Crossrefs

Differences are A000126.

Second differences are A000071 (Fibonacci -1).

Cf. A000045.

Sequence in context: A000127 A133552 A174439 * A106399 A334636 A299026

Adjacent sequences: A000125 A000126 A000127 * A000129 A000130 A000131

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Michel ten Voorde, Oct 06 2002

Status

approved