A005689 Number of Twopins positions. (Formerly M1042)

1, 2, 4, 7, 11, 16, 22, 30, 42, 61, 91, 137, 205, 303, 443, 644, 936, 1365, 1999, 2936, 4316, 6340, 9300, 13625, 19949, 29209, 42785, 62701, 91917, 134758, 197548, 289547

Offset

6,2

References

R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=6..37.

R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.

R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]

V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,2).

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 (2,-1,0,0,0,1).

Formula

G.f.: x^6*(1+x^2+x^3+x^4+x^5)/(1-2x+x^2-x^6). - Ralf Stephan, Apr 20 2004

Sum{k=0..floor(n/6), binomial(n-4k, 2k)} is 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, ... - Paul Barry, Sep 16 2004

Maple

A005689:=-(1+z**2+z**3+z**4+z**5)/(z**3+z-1)/(z**3-z+1); [Conjectured by Simon Plouffe in his 1992 dissertation.]

Mathematica

LinearRecurrence[{2, -1, 0, 0, 0, 1}, {1, 2, 4, 7, 11, 16}, 40] (* Harvey P. Dale, Feb 02 2019 *)

Crossrefs

Sequence in context: A334251 A175776 A177176 * A212365 A131075 A365698

Adjacent sequences: A005686 A005687 A005688 * A005690 A005691 A005692

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved