A005685 Number of Twopins positions.

1, 2, 3, 5, 7, 11, 16, 26, 40, 65, 101, 163, 257, 416, 663, 1073, 1719, 2781, 4472, 7236, 11664, 18873, 30465, 49293, 79641, 128862, 208315, 337061, 545071, 881943, 1426520, 2308158, 3733880, 6041545, 9774133

Offset: 4

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 and a(3) = 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]
• 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

Programs

Maple

A005685 := -(-1-z**3+2*z**4+z**2+z**5+z**6+z**7)/(z**2-z+1)/(z**2+z-1)/(z**4+z**2-1);

PARI

a(n)=(2*fibonacci(floor((n+1)/2))+fibonacci(n)+[0,1,1,0,-1,-1][(n%6)+1])/4; /* Ralf Stephan, Aug 25 2013 */

Formula

G.f.: (-x^4*(x^7+x^6+x^5+2*x^4-x^3+x^2-1))/((x^4+x^2-1)*(x^2-x+1)*(x^2+x-1)). - Conjectured by Simon Plouffe in his 1992 dissertation.

a(n) = Sum_{k=0..floor((n-1)/2)} A102541(n-1, 2*k), n >= 4. - Johannes W. Meijer, Aug 24 2013

a(n) = (1/4) * (2*F(floor((n+1)/2)) + F(n) + A010892(n-1)), with F(n) = A000045(n) the Fibonacci numbers. - Ralf Stephan, from Plouffe's g.f. Aug 25 2013

Extensions

More terms from Johannes W. Meijer, Aug 24 2013