A005688 Numbers of Twopins positions.

1, 2, 3, 5, 7, 10, 14, 20, 30, 45, 69, 104, 157, 236, 356, 540, 821, 1252, 1908, 2909, 4434, 6762, 10319, 15755, 24066, 36766, 56176, 85837, 131172, 200471, 306410, 468371, 715975, 1094516, 1673232, 2557997, 3910683

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 and a(4) =1. The formula for a(n) confirms these values. - Johannes W. Meijer, Aug 24 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

• 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]
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1,2,-2,0,0,0,-1).

Programs

Mathematica

LinearRecurrence[{2,0,-2,1,2,-2,0,0,0,-1},{1,2,3,5,7,10,14,20,30,45},40] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: (x^5*(1-x^2+x^3-2*x^5-x^6-x^7-x^8-x^9))/((1-x^2-x^5)*(1-2*x+x^2-x^5)). - Ralf Stephan, Apr 22 2004

a(n) = sum(A102541(n-k-1, 2*k), k=0..floor((n-1)/3)), n >= 5. - Johannes W. Meijer, Aug 24 2013

Extensions

More terms from Johannes W. Meijer, Aug 24 2013