A005676 a(n) = Sum_{k=0..n} C(n-k,4*k). (Formerly M1610)

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

0,6

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).

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)

Maple

A005676:=(z-1)**3/(-1+4*z-6*z**2+4*z**3-z**4+z**5); # Simon Plouffe in his 1992 dissertation.

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 *)

Prog

(Magma) [&+[Binomial(n-k, 4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, Sep 08 2017

Crossrefs

Column k=4 of A306680.

Sequence in context: A145126 A348289 A365733 * A365735 A365079 A038503

Adjacent sequences: A005673 A005674 A005675 * A005677 A005678 A005679

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from James A. Sellers, Aug 21 2000

Status

approved