A373694 Number of incongruent n-sided periodic Reinhardt polygons.

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 5, 0, 1, 5, 1, 2, 10, 1, 1, 12, 4, 1, 23, 2, 1, 38, 1, 0, 64, 1, 12, 102, 1, 1, 191, 12, 1, 329, 1, 2, 633, 1, 1, 1088, 9, 34, 2057, 2, 1, 3771, 66, 12, 7156, 1, 1, 13464, 1, 1, 25503, 0, 193, 48179, 1, 2, 92206, 358, 1, 175792, 1, 1, 338202

Offset

1,9

Links

Table of n, a(n) for n=1..75.

Kevin G. Hare and Michael J. Mossinghoff, Sporadic Reinhardt Polygons, Discrete & Computational Geometry. An International Journal of Mathematics and Computer Science 49, no. 3 (2013): 540-57.

Kevin G. Hare and Michael J. Mossinghoff, Most Reinhardt Polygons Are Sporadic, Geom. Dedicata 198 (2019): 1-18.

Michael J. Mossinghoff, Enumerating Isodiametric and Isoperimetric Polygons, J. Combin. Theory Ser. A 118, no. 6 (2011): 1801-15.

Michael Mossinghoff, I love Reinhardt Polygons, ICERM 2014.

Wikipedia, Reinhardt polygon

Formula

a(n) = A374832(n) - A373695(n).

a(n) = Sum_{d|n, d>1} D(n/d)*Mu(2d), with D(m) = 2^floor((m-3)/2) + (Sum_{d|m, d odd} 2^(m/d)*Phi(d) )/(4m), where Mu is MoebiusMu and Phi is EulerPhi.

Mathematica

dD[m_] := 2^Floor[(m - 3)/2] + Sum[2^(m/d) EulerPhi[d], {d, DeleteCases[Divisors[m], _?EvenQ]}]/4/m;
a[n_] := Sum[dD[n/d] MoebiusMu[2 d], {d, DeleteCases[Divisors[n], 1]}];

Crossrefs

Cf. A000010, A008683, A374832, A373695.

Sequence in context: A210705 A291771 A186972 * A374832 A053734 A238904

Adjacent sequences: A373691 A373692 A373693 * A373695 A373696 A373697

Keyword

nonn

Author

Bernd Mulansky, Aug 04 2024

Status

approved