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A005685 Number of Twopins positions.

%I A005685 M0664 #46 Jul 08 2025 16:33:30
%S A005685 1,2,3,5,7,11,16,26,40,65,101,163,257,416,663,1073,1719,2781,4472,
%T A005685 7236,11664,18873,30465,49293,79641,128862,208315,337061,545071,
%U A005685 881943,1426520,2308158,3733880,6041545,9774133
%N A005685 Number of Twopins positions.
%C A005685 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
%D A005685 R. K. Guy, ``Anyone for Twopins?,'' in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
%D A005685 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005685 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]
%H A005685 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005685 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%F A005685 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.
%F A005685 a(n) = Sum_{k=0..floor((n-1)/2)} A102541(n-1, 2*k), n >= 4. - _Johannes W. Meijer_, Aug 24 2013
%F A005685 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
%p A005685 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);
%o A005685 (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 */
%K A005685 nonn
%O A005685 4,2
%A A005685 _N. J. A. Sloane_
%E A005685 More terms from _Johannes W. Meijer_, Aug 24 2013