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A321416
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Number of n element multisets of the 10th roots of unity with zero sum.
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2
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1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140, 360, 254, 565, 430, 855, 690, 1255, 1060, 1795, 1570, 2510, 2256, 3440, 3160, 4630, 4330, 6132, 5820, 8005, 7690, 10315, 10008, 13135, 12850, 16545, 16300, 20634, 20450, 25500, 25400, 31250, 31260
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OFFSET
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0,3
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COMMENTS
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Equivalently, the number of closed convex paths of length n whose steps are the 10th roots of unity up to translation. For even n, there will be 5 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,7,3,-8,3,7,-6,-4,4,1,-1)
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FORMULA
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G.f.: (1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2).
G.f.: (1 - x + x^2 - x^3 + x^4)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^6*(1 + x)^4).
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MATHEMATICA
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LinearRecurrence[{1, 4, -4, -6, 7, 3, -8, 3, 7, -6, -4, 4, 1, -1}, {1, 0, 5, 0, 15, 2, 35, 10, 70, 30, 128, 70, 220, 140}, 50] (* Jinyuan Wang, Feb 28 2020 *)
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PROG
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(PARI) Vec((1 - x^10)/((1 - x^2)^5 * (1 - x^5)^2) + O(x^50))
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CROSSREFS
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Column k=5 of A321414.
Cf. A053090, A070190.
Sequence in context: A290867 A027635 A291218 * A226372 A093782 A085105
Adjacent sequences: A321413 A321414 A321415 * A321417 A321418 A321419
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KEYWORD
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nonn,easy
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AUTHOR
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Andrew Howroyd, Nov 09 2018
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STATUS
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approved
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