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A331143
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Number of coincidence site modules of icosian ring of index n.
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1
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1, 0, 0, 25, 36, 0, 0, 0, 100, 0, 288, 0, 0, 0, 0, 410, 0, 0, 800, 900, 0, 0, 0, 0, 912, 0, 0, 0, 1800, 0, 2048, 0, 0, 0, 0, 2500, 0, 0, 0, 0, 3528, 0, 0, 7200, 3600, 0, 0, 0, 2500, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 7200, 0, 7688, 0, 0, 6600, 0, 0, 0, 0, 0, 0
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OFFSET
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1,4
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LINKS
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Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See Section 4. Caution: there is a typo in a(19) here and in other papers.
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FORMULA
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See Zeiner (2015) for the formula and the Dirichlet g.f. (but beware of the typo in the 19th term).
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MATHEMATICA
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h[x_, 0] := 1;
h[x_, r_] := (x^(2 r + 1) + x^(2 r - 2) - 2 x^Quotient[r - 1, 2] If[EvenQ[r], (1 + x^2)/(1 + x), 1]) (x + 1)^2/(x^3 - 1);
apr[5, r_] := h[5, r];
apr[p_?(Abs@Mod[#, 5, -1] == 1 &), r_] := Sum[h[p, r - s] h[p, s], {s, 0, r}];
apr[p_, r_] := If[OddQ[r], 0, h[p^2, r/2]];
a[1] = 1;
a[n_] := Product[apr @@ pr, {pr, FactorInteger[n]}];
Table[a[n], {n, 100}]
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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