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

1,4

Links

Table of n, a(n) for n=1..70.

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.

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

Adjacent sequences: A331140 A331141 A331142 * A331144 A331145 A331146

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