A001636 A Fielder sequence: a(n) = a(n-1) + a(n-2) - a(n-7), n >= 8. (Formerly M0763 N0290)

0, 2, 3, 6, 10, 17, 21, 38, 57, 92, 143, 225, 351, 555, 868, 1366, 2142, 3365, 5282, 8296, 13023, 20451, 32108, 50417, 79160, 124295, 195159, 306431, 481139, 755462, 1186184, 1862486, 2924375, 4591702, 7209646, 11320209, 17774393, 27908418, 43820325

Offset

1,2

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, 1, 0, 0, 0, 0, -1).

Formula

G.f.: x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7).

a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), n >= 7.

Maple

A001636:=-z*(2+3*z+4*z**2+5*z**3+6*z**4)/(z+1)/(z**5+z**3+z-1); # Simon Plouffe in his 1992 dissertation
a:= n -> (Matrix([[6, -1$4, 4, 5]]). Matrix(7, (i, j)-> if (i=j-1) then 1 elif j=1 then [1$2, 0$4, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=1..38); # Alois P. Heinz, Aug 01 2008

Mathematica

LinearRecurrence[{1, 1, 0, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 17, 21}, 50] (* T. D. Noe, Aug 09 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, polcoeff(x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7)+x*O(x^n), n))
(Magma) I:=[0, 2, 3, 6, 10, 17, 21]; [n le 7 select I[n] else Self(n-1) + Self(n-2) - Self(n-7): n in [1..30]]; // G. C. Greubel, Jan 09 2018

Crossrefs

Cf. A013983.

Sequence in context: A066895 A105075 A140669 * A347786 A036588 A334893

Adjacent sequences: A001633 A001634 A001635 * A001637 A001638 A001639

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Feb 17 2002

Status

approved