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%I M0643 #22 Nov 10 2017 14:10:26
%S 1,2,3,5,7,10,13,18,24,35,50,75,109,161,231,336,482,703,1020,1498,
%T 2188,3214,4694,6877,10039,14699,21487,31489,46097,67582,98977,145071,
%U 212463,311344,456045,668328,979182,1435107,2102900,3082037,4516347,6618985,9699527,14215176
%N Number of Twopins positions.
%C 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
%D R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005691/b005691.txt">Table of n, a(n) for n = 6..1000</a>
%H R. K. Guy, <a href="/A005251/a005251_1.pdf">Anyone for Twopins?</a>, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
%F 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
%F a(n) = Sum_{k=0..floor((n-1)/2)} A228570(n-1, 2*k), n >= 6. - _Johannes W. Meijer_, Aug 26 2013
%t 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 *)
%Y Cf. A228570.
%K nonn
%O 6,2
%A _N. J. A. Sloane_
%E Extended by _Johannes W. Meijer_, Aug 26 2013
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