A005682 Number of Twopins positions.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• 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).

Programs

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 *)

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

Extensions

More terms from David W. Wilson