A005691 Number of Twopins positions. (Formerly M0643)

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

6,2

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]

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

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 *)

Crossrefs

Cf. A228570.

Sequence in context: A038083 A238863 A060688 * A035954 A023192 A064480

Adjacent sequences: A005688 A005689 A005690 * A005692 A005693 A005694

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by Johannes W. Meijer, Aug 26 2013

Status

approved