A005691 Number of Twopins positions.

1, 2, 3, 5, 7, 10, 13, 18, 24, 35, 50, 75, 109, 161, 231, 336, 482, 703, 1020, 1498, 2188, 3214, 4694, 6877, 10039, 14699, 21487, 31489, 46097, 67582, 98977, 145071, 212463, 311344, 456045, 668328, 979182, 1435107, 2102900, 3082037, 4516347, 6618985, 9699527, 14215176

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 1 and a(5) = 1. The formula for a(n) confirms these values. - Johannes W. Meijer, Aug 26 2013

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

• Vincenzo Librandi, Table of n, a(n) for n = 6..1000
• 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]

Crossrefs

Cf. A228570.

Programs

Mathematica

CoefficientList[Series[((1 - x^2 + x^3 - 2*x^6 - x^7 - x^8 - x^9 - x^10 - x^11))/((x^3 - x + 1) (x^3 + x - 1) (x^6 + x^2 - 1)), {x, 0, 50}], x] (* Wesley Ivan Hurt, May 03 2017 *)

Formula

G.f.: (x^6*(1-x^2+x^3-2*x^6-x^7-x^8-x^9-x^10-x^11))/((x^3-x+1)*(x^3+x-1)*(x^6+x^2-1)). - Ralf Stephan, Apr 22 2004

a(n) = Sum_{k=0..floor((n-1)/2)} A228570(n-1, 2*k), n >= 6. - Johannes W. Meijer, Aug 26 2013

Extensions

Extended by Johannes W. Meijer, Aug 26 2013