A374832 -id:A374832 - cp's OEIS frontend

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

Bernd Mulansky, Aug 04 2024

nonn

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

Cf. A000010, A008683, A374832, A373695.

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]}];

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.

A373695 Number of incongruent n-sided "sporadic" Reinhardt polygons.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 144, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4392, 0, 0, 1308, 0, 0, 93, 0, 0, 0, 27, 0, 0, 0, 0, 153660, 0, 0, 315, 0, 0, 0, 0, 0, 161028, 0, 0, 0, 0

Offset: 1

Bernd Mulansky, Aug 04 2024

nonn

The first nonzero entries are a(30)=3, a(42)=9, a(45)=144, a(60)=4392. It is proved that a(2^a p^b)=0, if p is an odd prime, a,b>=0. Also a(pq)=0 and a(2pq)=(2^(p-1)-1)(2^(q-1)-1)/(pq), if p and q are distinct odd primes.

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

Cf. A374832, A373694.

a(n) = A374832(n) - A373694(n).

cp's OEIS Frontend

A373694 Number of incongruent n-sided periodic Reinhardt polygons.

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