OFFSET
1,3
COMMENTS
For n>=4 also number of Napier cycle types for dimension d=n-3. See Böhm link. - Hugo Pfoertner, Oct 01 2013
Also the number of combinatorial types of simplicial neighborly polytopes in dimension 2n - 3 with 2n vertices. This sequence was described before the enumeration of self-dual necklaces: see references. See links for a bijection between the two objects. - Moritz Firsching, Aug 13 2015
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Oswin Aichholzer and Anna Brötzner, Bicolored Order Types, Comp. Geom. Topology (2024) Vol. 3, No. 2, 3:1-3:17.
Amos Altshuler and Peter McMullen, The number of simplicial neighbourly d-polytopes with d + 3 vertices, Mathematika, 20(02):263-266, 1973., Theorem 1, p. 263.
Johannes Böhm, Generalized hyperbolic Napier cycles and their hyperbolic kernels, Part III, Jenaer Schriften zur Mathematik und Informatik, Math/inf/06/08, 2008.
Moritz Firsching, Realizability and inscribability for some simplicial spheres and matroid polytopes, arXiv:1508.02531v1 [math.MG], 2015. See Appendix A1.
Bernd Mulansky and Andreas Potschka, A zonogon approach for computing small polygons of maximum perimeter, arXiv:2404.01841 [math.OC], 2024. See p. 9. See also Math. Program., 2025.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
Zhe Sun, Takuya Suenaga, Parantap Sarkar, Sota Sato, Motoko Kotani, and Hiroyuki Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Natl. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114.
FORMULA
a(n) = A263768(n) + 1. - Bernd Mulansky, Mar 08 2023
MATHEMATICA
a[n_] := (1/2)*(2^Quotient[n-1, 2] + Total[(Mod[#, 2]*EulerPhi[#]*2^(n/#) & ) /@ Divisors[n]]/(2*n)); Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 24 2011, after Pari *)
PROG
(PARI) a(n)= (1/2) *(2^((n-1)\2)+sumdiv(n, k, (k%2)*eulerphi(k)*2^(n/k))/(2*n))
(SageMath)
def a(n):
return 2^floor((n-3)/2)+1/(4*(n))*sum([euler_phi(h)*2^((n)/h) for h in divisors(n) if is_odd(h)])
# Moritz Firsching, Aug 13 2015
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Michael Somos
STATUS
approved