A005676 a(n) = Sum_{k=0..n} C(n-k,4*k).
1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 128, 220, 376, 661, 1211, 2290, 4382, 8347, 15706, 29191, 53824, 99009, 182497, 337745, 627401, 1167937, 2174834, 4046070, 7517368, 13951852, 25880583, 48009456, 89090436, 165392856, 307137901
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,4).
- V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
- 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,-6,4,-1,1).
Crossrefs
Column k=4 of A306680.
Programs
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Magma
[&+[Binomial(n-k, 4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, Sep 08 2017
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Maple
A005676:=(z-1)**3/(-1+4*z-6*z**2+4*z**3-z**4+z**5); # Simon Plouffe in his 1992 dissertation.
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Mathematica
LinearRecurrence[{4, -6, 4, -1, 1}, {1, 1, 1, 1, 1}, 40] (* or *) CoefficientList[Series[(1 - x)^3 / ((1 - x)^4 - x^5), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 08 2017 *)
Formula
From Paul Barry, Jul 23 2004: (Start)
G.f.: (1-3x+3x^2-x^3)/(1-4x+6x^2-4x^3+x^4-x^5) = (1-x)^3/((1-x)^4-x^5).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, 4k).
a(n) = 4a(n-1)-6a(n-2)+4a(n-3)-a(n-4)+a(n-5). (End)
Extensions
More terms from James Sellers, Aug 21 2000