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A007147 Number of self-dual 2-colored necklaces with 2n beads.
(Formerly M0319)
4
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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

Sequence in context: A102526 A050192 A191786 * A230380 A127968 A188541

Adjacent sequences:  A007144 A007145 A007146 * A007148 A007149 A007150

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos

STATUS

approved

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Last modified November 20 08:16 EST 2018. Contains 317385 sequences. (Running on oeis4.)