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).
Zhe Sun, T Suenaga, P Sarkar, S Sato, M Kotani, H Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nath. Acead. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113
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:1508.02531 [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.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
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))
(Sage)
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