A005682 Number of Twopins positions. (Formerly M1106)

1, 2, 4, 8, 15, 28, 51, 92, 165, 294, 522, 924, 1632, 2878, 5069, 8920, 15686, 27570, 48439, 85080, 149405, 262320, 460515, 808380, 1418916, 2490432, 4370944, 7671188, 13462945, 23627078, 41464296, 72766972, 127700055, 224101844, 393276447, 690158844, 1211153337

Offset

5,2

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=5..41.

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]

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

Formula

a(n) = 2a(n-1) - a(n-4) - a(n-6). - John W. Layman

G.f.: x^5/((x^3+x^2-1)*(x^3-x^2+2*x-1)). - Ralf Stephan, Apr 22 2004

a(n) = (A005251(n+1)-A000931(n+5))/2. - R. J. Mathar, Dec 10 2011

Maple

A005682:=1/((z**3-z**2+2*z-1)*(z**3+z**2-1)); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation for offset 0

Mathematica

CoefficientList[ Series[x^5/((x^3 + x^2 - 1)(x^3 - x^2 + 2 x - 1)), {x, 0, 41}], x] (* or *)
a[n_] := a[n] = 2 a[n - 1] - a[n - 4] - a[n - 6]; a[0] = a[1] = a[2] = a[3] = a[4] = 0; a[5] = 1; Array[a, 42, 0] (* or *)
LinearRecurrence[{2, 0, 0, -1, 0, -1}, {0, 0, 0, 0, 0, 1}, 38] (* Robert G. Wilson v, Jun 22 2014 *)

Crossrefs

Sequence in context: A334635 A358836 A029907 * A114833 A065617 A062065

Adjacent sequences: A005679 A005680 A005681 * A005683 A005684 A005685

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved