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A000128 A nonlinear binomial sum.
(Formerly M1120 N0428)
1
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 (list; graph; refs; listen; history; text; internal format)
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
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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Michel ten Voorde, Oct 06 2002
STATUS
approved

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Last modified July 17 09:00 EDT 2024. Contains 374363 sequences. (Running on oeis4.)