A007147 Number of self-dual 2-colored necklaces with 2n beads. (Formerly M0319)

1, 1, 2, 2, 4, 5, 9, 12, 23, 34, 63, 102, 190, 325, 612, 1088, 2056, 3771, 7155, 13364, 25482, 48175, 92205, 175792, 337594, 647326, 1246863, 2400842, 4636390, 8956060, 17334801, 33570816, 65108062, 126355336, 245492244, 477284182

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.

Index entries for sequences related to necklaces

Formula

a(n) = (1/2) * (A016116(n-1) + A000016(n)).

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

Cf. A000016, A016116, A263768.

Sequence in context: A102526 A050192 A191786 * A230380 A127968 A330627

Adjacent sequences: A007144 A007145 A007146 * A007148 A007149 A007150

Keyword

nonn,nice,easy,changed

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos.

Status

approved