A005682 - OEIS

%I M1106 #47 Apr 13 2022 13:25:17

%S 1,2,4,8,15,28,51,92,165,294,522,924,1632,2878,5069,8920,15686,27570,

%T 48439,85080,149405,262320,460515,808380,1418916,2490432,4370944,

%U 7671188,13462945,23627078,41464296,72766972,127700055,224101844,393276447,690158844,1211153337

%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. 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 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 (2,0,0,-1,0,-1).

%F a(n) = 2a(n-1) - a(n-4) - a(n-6). - _John W. Layman_

%F G.f.: x^5/((x^3+x^2-1)*(x^3-x^2+2*x-1)). - _Ralf Stephan_, Apr 22 2004

%F a(n) = (A005251(n+1)-A000931(n+5))/2. - _R. J. Mathar_, Dec 10 2011

%p A005682:=1/((z**3-z**2+2*z-1)*(z**3+z**2-1)); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation for offset 0

%t CoefficientList[ Series[x^5/((x^3 + x^2 - 1)(x^3 - x^2 + 2 x - 1)), {x, 0, 41}], x] (* or *)

%t a[n_] := a[n] = 2 a[n - 1] - a[n - 4] - a[n - 6]; a[0] = a[1] = a[2] = a[3] = a[4] = 0; a[5] = 1; Array[a, 42, 0] (* or *)

%t LinearRecurrence[{2, 0, 0, -1, 0, -1}, {0, 0, 0, 0, 0, 1}, 38] (* _Robert G. Wilson v_, Jun 22 2014 *)

%K nonn,easy

%O 5,2

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_