A001639 A Fielder sequence. a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5), n >= 6. (Formerly M3353 N1349)

1, 1, 4, 9, 16, 22, 36, 65, 112, 186, 309, 522, 885, 1492, 2509, 4225, 7124, 12010, 20236, 34094, 57453, 96823, 163163, 274946, 463316, 780755, 1315687, 2217112, 3736129, 6295887, 10609441, 17878369, 30127497, 50768954, 85552651, 144167958, 242942778

Offset

1,3

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 = 1..1000

Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.

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 (1, 0, 1, 1, 1).

Formula

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

Maple

A001639:=-(1+3*z**2+4*z**3+5*z**4)/(-1+z+z**3+z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Drop[CoefficientList[Series[x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5), {x, 0, 40}], x], 1] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{1, 0, 1, 1, 1}, {1, 1, 4, 9, 16}, 30] (* G. C. Greubel, Jan 09 2018 *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5)+x*O(x^n), n))
(Magma) I:=[1, 1, 4, 9, 16]; [n le 5 select I[n] else Self(n-1) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 09 2018

Crossrefs

Cf. A000570.

Sequence in context: A313350 A152399 A022822 * A309138 A162207 A092614

Adjacent sequences: A001636 A001637 A001638 * A001640 A001641 A001642

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Feb 17 2002

Status

approved