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A005686
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Number of Twopins positions.
(Formerly M0267)
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4
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0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, 18, 22, 27, 34, 41, 52, 63, 79, 97, 120, 149, 183, 228, 280, 348, 429, 531, 657, 811, 1005, 1240, 1536, 1897, 2347, 2902, 3587, 4438, 5484, 6785, 8386, 10372, 12824, 15856, 19609, 24242, 29981, 37066, 45837
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OFFSET
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0,7
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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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a(n) = Sum_{k=0..floor(n/2)} binomial(floor((n+3k-3)/5), k). - Paul Barry, Jul 10 2004
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MAPLE
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A005686 := -(z+1)*(z**3+z+1)/(-1+z**2+z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 1's
a := proc(n): if n = 0 then 0 else add(binomial(floor((n+3*k-4)/5), k), k=0..floor((n-1)/2)) fi: end: seq(a(n), n=0..54); # Johannes W. Meijer, Aug 05 2013
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MATHEMATICA
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nn=54; CoefficientList[Series[(x+x^2)/(1-x^2-x^5), {x, 0, nn}], x] (* Geoffrey Critzer, Apr 28 2013 *)
m = 5; For[n = 0, n < m, n++, a[n] = 1]; For[n = m, n < 51, n++, a[n] = a[n - m] + a[n - 2]]; Table[a[n], {n, 0, 50}] (*Sergio Falcon, Nov 12 2015 *)
Join[{0}, LinearRecurrence[{0, 1, 0, 0, 1}, {1, 1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)
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PROG
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(PARI) a(n)=if(n<0, polcoeff((x^3+x^4)/(1+x^3-x^5)+x^-n*O(x), -n), polcoeff((x+x^2)/(1-x^2-x^5)+x^n*O(x), n)) /* Michael Somos, Jul 15 2004 */
(PARI) a(n)=sum(k=0, (n-1)\2, binomial((n+3*k-4)\5, k))
(Magma) I:=[1, 1, 1, 1, 1]; [0] cat [n le 5 select I[n] else Self(n-2)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jan 19 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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