A005689 - OEIS

%I M1042 #41 Jan 05 2025 19:51:33

%S 1,2,4,7,11,16,22,30,42,61,91,137,205,303,443,644,936,1365,1999,2936,

%T 4316,6340,9300,13625,19949,29209,42785,62701,91917,134758,197548,

%U 289547

%N Number of Twopins positions.

%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 R. Austin and R. K. Guy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-1/austin.pdf">Binary sequences without isolated ones</a>, Fib. Quart., 16 (1978), 84-86.

%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]

%H V. C. Harris and C. C. Styles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/harris.pdf">A generalization of Fibonacci numbers</a>, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,2).

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1).

%F G.f.: x^6*(1+x^2+x^3+x^4+x^5)/(1-2x+x^2-x^6). - _Ralf Stephan_, Apr 20 2004

%F Sum{k=0..floor(n/6), binomial(n-4k, 2k)} is 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, ... - _Paul Barry_, Sep 16 2004

%p A005689:=-(1+z**2+z**3+z**4+z**5)/(z**3+z-1)/(z**3-z+1); [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%t LinearRecurrence[{2,-1,0,0,0,1},{1,2,4,7,11,16},40] (* _Harvey P. Dale_, Feb 02 2019 *)

%K nonn

%O 6,2

%A _N. J. A. Sloane_.