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A331143 Number of coincidence site modules of icosian ring of index n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
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.
Michael Baake and Peter Zeiner, Geometric Enumeration Problems for Lattices and Embedded Z-Modules, in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172; arXiv:1709.07317 [math.MG], 2017. See Theorem 3.11.12 (or Theorem 11.12 in the arXiv version).
FORMULA
See Zeiner (2015) for the formula and the Dirichlet g.f. (but beware of the typo in the 19th term).
MATHEMATICA
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}]
(* Andrey Zabolotskiy, Feb 16 2021 *)
CROSSREFS
Cf. A031366.
Sequence in context: A153446 A039457 A243749 * A031366 A068165 A086935
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Jan 12 2020
EXTENSIONS
New name, a(19) corrected, a(29) and beyond added by Andrey Zabolotskiy, Feb 16 2021
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)