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A331143 Number of coincidence site modules of icosian ring of index n.

%I A331143 #24 Feb 16 2021 13:14:23
%S A331143 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,
%T A331143 0,1800,0,2048,0,0,0,0,2500,0,0,0,0,3528,0,0,7200,3600,0,0,0,2500,0,0,
%U A331143 0,0,0,10368,0,0,0,7200,0,7688,0,0,6600,0,0,0,0,0,0
%N A331143 Number of coincidence site modules of icosian ring of index n.
%H A331143 Michael Baake and Peter Zeiner, <a href="https://doi.org/10.1080/14786430701846206">Coincidences in 4 dimensions</a>, Phil. Mag. 88 (2008), 2025-2032; arXiv:<a href="https://arxiv.org/abs/0712.0363">0712.0363</a> [math.MG]. See Section 4. Caution: there is a typo in a(19) here and in other papers.
%H A331143 Michael Baake and Peter Zeiner, <a href="https://doi.org/10.1017/9781139033862.005">Geometric Enumeration Problems for Lattices and Embedded Z-Modules</a>, in: <a href="http://www.aperiodicorder.org/">Aperiodic Order</a>, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172; arXiv:<a href="https://arxiv.org/abs/1709.07317">1709.07317</a> [math.MG], 2017. See Theorem 3.11.12 (or Theorem 11.12 in the arXiv version).
%H A331143 Peter Zeiner, <a href="https://core.ac.uk/display/211846842">Coincidence Site Lattices and Coincidence Site Modules</a>, 2015. See p. 83.
%F A331143 See Zeiner (2015) for the formula and the Dirichlet g.f. (but beware of the typo in the 19th term).
%t A331143 h[x_, 0] := 1;
%t A331143 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);
%t A331143 apr[5, r_] := h[5, r];
%t A331143 apr[p_?(Abs@Mod[#, 5, -1] == 1 &), r_] := Sum[h[p, r - s] h[p, s], {s, 0, r}];
%t A331143 apr[p_, r_] := If[OddQ[r], 0, h[p^2, r/2]];
%t A331143 a[1] = 1;
%t A331143 a[n_] := Product[apr @@ pr, {pr, FactorInteger[n]}];
%t A331143 Table[a[n], {n, 100}]
%t A331143 (* _Andrey Zabolotskiy_, Feb 16 2021 *)
%Y A331143 Cf. A031366.
%K A331143 nonn,mult
%O A331143 1,4
%A A331143 _N. J. A. Sloane_, Jan 12 2020
%E A331143 New name, a(19) corrected, a(29) and beyond added by _Andrey Zabolotskiy_, Feb 16 2021