A001584 A generalized Fibonacci sequence.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 8, 12, 12, 16, 21, 21, 31, 37, 38, 58, 65, 71, 106, 114, 135, 191, 201, 257, 341, 359, 485, 605, 652, 904, 1070, 1202, 1664, 1894, 2237, 3029, 3370, 4176, 5464, 6048, 7779, 9793, 10963, 14411, 17492, 20054, 26507, 31239, 36924, 48396

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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 = 0..1000
• V. C. Harris and C. C. Styles, Generalized Fibonacci sequences associated with a generalized Pascal triangle, Fib. Quart., 4 (1966), 241-248.
• V. C. Harris and C. C. Styles, Generalized Fibonacci sequences associated with a generalized Pascal triangle and accompanying letter (annotated scanned copy)
• Alaa Ibrahim and Bruno Salvy, Positivity Proofs for Linear Recurrences through Contracted Cones, arXiv:2412.08576 [cs.SC], 2024. See p. 22.
• 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 (0, 0, 2, 0, 0, -1, 0, 1).

Crossrefs

Cf. A017817.

Programs

Maple

A001584:=(z-1)*(z**2+z+1)**2/(z**4-z**3+1)/(z**4+z**3-1); # Simon Plouffe in his 1992 dissertation

PARI

Vec((1+x+x^2-x^3-x^4-x^5)/(1-2*x^3+x^6-x^8) + O(x^80)) \\ Michel Marcus, Sep 07 2017

Formula

G.f.: (1 + x + x^2 - x^3 - x^4 - x^5)/(1 - 2*x^3 + x^6 - x^8).

Extensions

More terms from David W. Wilson