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A005683
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Numbers of Twopins positions.
(Formerly M0695)
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5
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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
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OFFSET
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3,2
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COMMENTS
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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
a(n) is the number of equivalence classes of subsets of {1..n-2} without isolated elements up to reflection. The reflection of a subset is the set obtained by mapping each element i to n + 1 - i. For example, the a(6)=5 equivalence classes of subsets of {1..4} are {}, {1,2}/{3,4}, {2,3}, {1,2,3}/{2,3,4}, {1,2,3,4}. If reflections are not considered equivalent then A005251(n) gives the number of subsets of {1..n-2} without isolated elements. - Andrew Howroyd, Dec 24 2019
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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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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]
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FORMULA
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G.f.: x^3*(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
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MAPLE
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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.]
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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