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

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

Offset: 1

Author

N. J. A. Sloane, Jan 12 2020

Keywords

• nonn
• mult

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).
• Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, 2015. See p. 83.

Crossrefs

Cf. A031366.

Programs

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 *)

Formula

See Zeiner (2015) for the formula and the Dirichlet g.f. (but beware of the typo in the 19th term).

Extensions

New name, a(19) corrected, a(29) and beyond added by Andrey Zabolotskiy, Feb 16 2021