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

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).

A373694 Number of incongruent n-sided periodic Reinhardt polygons.

Original entry on oeis.org

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.

A374832 Number of incongruent n-sided Reinhardt polygons.

Original entry on oeis.org

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, 41, 1, 0, 64, 1, 12, 102, 1, 1, 191, 12, 1, 338, 1, 2, 777, 1, 1, 1088, 9, 34, 2057, 2, 1, 3771, 66, 12, 7156, 1, 1, 17856, 1, 1, 26811, 0, 193, 48272, 1, 2, 92206, 385, 1, 175792

Offset: 1

Bernd Mulansky, Jul 21 2024

nonn

Karl Reinhardt, Extremale Polygone gegebenen Durchmessers. Jahresber. Deutsche Math.-Verein. 31 (1922): 251-70.

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.
Wikipedia, Reinhardt polygon.

Cf. A373694, A373695.

a(n) = A373694(n) + A373695(n). - Bernd Mulansky, Aug 23 2024

More terms from Bernd Mulansky, Aug 23 2024

A345731 Additive bases: a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of distinct elements) of which are distinct.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 24, 34, 45, 57, 71, 86, 105, 126, 148

Offset: 2

Bernd Mulansky, Jun 25 2021

Such sets are known as weak Sidon sets, weak B_2 sets, or well-spread sequences.

n - 1 <= a(n) <= A003022(n). - Michael S. Branicky, Jun 25 2021

			a(6)=12 because 0-1-2-4-7-12 (0-5-8-10-11-12) resp. 0-1-2-6-9-12 (0-3-6-10-11-12) are shortest weak Sidon sets of size 6.
a(16)=148: [0, 3, 5, 6, 32, 49, 59, 68, 93, 106, 118, 126, 130, 134, 141, 148]. - _Zhao Hui Du_, Jul 27 2025

Alison M. Marr and W. D. Wallis, Magic Graphs, Birkhäuser, 2nd ed., 2013. See Section 2.3.
Xiaodong Xu, Meilian Liang, and Zehui Shao, On weak Sidon sequences, The Journal of Combinatorial Mathematics and Combinatorial Computing (2014), 107--113

A. Lladó, Largest cliques in connected supermagic graphs, European Journal of Combinatorics, Vol. 28, No. 8 (2007), 2240-2247.

See A003022, A004133, and A004135 for other versions.

a[n_Integer?NonNegative] := Module[{k = n - 1}, While[SelectFirst[Subsets[Range[0, k - 1], {n - 1}], Length@Union[Plus @@@ Subsets[#~Join~{k}, {2}]] >= (n*(n - 1))/2 &] === Missing["NotFound"], k++]; k];
Table[a[n], {n, 2, 8}] (* Robert P. P. McKone, Nov 05 2023 *)

from itertools import combinations, count
def a(n):
    for k in count(n-1):
        for c in combinations(range(k), n-1):
            c = c + (k,)
            ss = set()
            for s in combinations(c, 2):
                if sum(s) in ss: break
                else: ss.add(sum(s))
            if len(ss) == n*(n-1)//2: return k # use (k, c) for sets
print([a(n) for n in range(2, 9)]) # Michael S. Branicky, Jun 25 2021

a(16) corrected and a(17) deleted by Zhao Hui Du, Jul 27 2025

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

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A373694 Number of incongruent n-sided periodic Reinhardt polygons.

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A374832 Number of incongruent n-sided Reinhardt polygons.

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A345731 Additive bases: a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of distinct elements) of which are distinct.

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