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A005689
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Number of Twopins positions.
(Formerly M1042)
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3
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1, 2, 4, 7, 11, 16, 22, 30, 42, 61, 91, 137, 205, 303, 443, 644, 936, 1365, 1999, 2936, 4316, 6340, 9300, 13625, 19949, 29209, 42785, 62701, 91917, 134758, 197548, 289547
(list;
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history;
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OFFSET
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6,2
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REFERENCES
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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).
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LINKS
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Table of n, a(n) for n=6..37.
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
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]
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,2).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1).
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FORMULA
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G.f.: x^6*(1+x^2+x^3+x^4+x^5)/(1-2x+x^2-x^6). - Ralf Stephan, Apr 20 2004
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
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MAPLE
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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.]
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MATHEMATICA
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LinearRecurrence[{2, -1, 0, 0, 0, 1}, {1, 2, 4, 7, 11, 16}, 40] (* Harvey P. Dale, Feb 02 2019 *)
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CROSSREFS
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Sequence in context: A334251 A175776 A177176 * A212365 A131075 A133523
Adjacent sequences: A005686 A005687 A005688 * A005690 A005691 A005692
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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