A005687 Number of Twopins positions.
1, 2, 4, 6, 9, 14, 22, 36, 57, 90, 139, 214, 329, 506, 780, 1200, 1845, 2830, 4337, 6642, 10170, 15572, 23838, 36486, 55828, 85408, 130641, 199814, 305599, 467366, 714735, 1092980, 1671335, 2555650, 3907781, 5975202, 9136288, 13969560, 21359528
Offset: 7
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, -2, 1, 2, -2, 0, 0, 0, -1).
Programs
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Maple
a:= n-> (Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [2,0,-2,1,2,-2,0,0,0,-1][i] else 0 fi)^n)[1,8]: seq(a(n), n=7..70); # Alois P. Heinz, Aug 14 2008
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Mathematica
LinearRecurrence[{2, 0, -2, 1, 2, -2, 0, 0, 0, -1}, {1, 2, 4, 6, 9, 14, 22, 36, 57, 90}, 40] (* Jean-François Alcover, Nov 12 2015 *)
Formula
G.f.: x^7/((1-x^2-x^5)*(1-2*x+x^2-x^5)). - Simon Plouffe in his 1992 dissertation.
Extensions
More terms from Alois P. Heinz, Aug 14 2008