A005685 Number of Twopins positions. (Formerly M0664)

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

4,2

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

Table of n, a(n) for n=4..38.

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

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(A102541(n-1, 2*k), k=0..floor((n-1)/2)), 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

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);

Prog

(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 */

Crossrefs

Sequence in context: A218022 A091980 A274113 * A141656 A092180 A050298

Adjacent sequences: A005682 A005683 A005684 * A005686 A005687 A005688

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Johannes W. Meijer, Aug 24 2013

Status

approved