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 A005683 Numbers of Twopins positions. (Formerly M0695) 3
 1, 2, 3, 5, 8, 13, 22, 37, 63, 108, 186, 322, 559, 973, 1697, 2964, 5183, 9071, 15886, 27835, 48790, 85545, 150021, 263136, 461596, 809812, 1420813, 2492945, 4374273, 7675598, 13468787, 23634817, 41474548, 72780553, 127718046, 224125677, 393308019, 690200668 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,2 COMMENTS Appears to be a bisection of A068930. - Ralf Stephan, Apr 20 2004 The Ze3 and Ze4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence with a(1) = 1 and a(2) = 1; the recurrence relation below confirms these values and gives a(0) = 0. - Johannes W. Meijer, Jul 14 2011 The complete sequence by R. K. Guy in "Anyone for Twopins?" starts with a(0)=0, a(1)=1 and a(2)=1 and has g.f. x*(1-x-x^2)/(1-2*x+x^4+x^6). - Johannes W. Meijer, Aug 14 2011 REFERENCES 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). LINKS 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] Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (2, 0, 0, -1, 0, -1). FORMULA G.f.: (1-x^2-x^3-x^4-x^5)/(1-2*x+x^4+x^6). - Ralf Stephan, Apr 20 2004 a(3)=1, a(4)=2, a(5)=3, a(6)=5, a(7)=8, a(8)=13, a(n)=2*a(n-1)- a(n-4)- a(n-6). - Harvey P. Dale, Jun 20 2011 MAPLE A005683:=-(-1+z**2+z**3+z**4+z**5)/(z**3-z**2+2*z-1)/(z**3+z**2-1); [Conjectured by Simon Plouffe in his 1992 dissertation.] MATHEMATICA CoefficientList[Series[(1-x^2-x^3-x^4-x^5)/(1-2x+x^4+x^6), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 0, 0, -1, 0, -1}, {1, 2, 3, 5, 8, 13}, 40] (* Harvey P. Dale, Jun 20 2011 *) CROSSREFS Sequence in context: A124429 A018152 A293078 * A173404 A213710 A288382 Adjacent sequences:  A005680 A005681 A005682 * A005684 A005685 A005686 KEYWORD nonn AUTHOR EXTENSIONS More terms from Harvey P. Dale, Jun 20 2011 STATUS approved

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Last modified November 18 18:05 EST 2018. Contains 317323 sequences. (Running on oeis4.)