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%I M0319 #69 Apr 11 2024 04:12:39
%S 1,1,2,2,4,5,9,12,23,34,63,102,190,325,612,1088,2056,3771,7155,13364,
%T 25482,48175,92205,175792,337594,647326,1246863,2400842,4636390,
%U 8956060,17334801,33570816,65108062,126355336,245492244,477284182
%N Number of self-dual 2-colored necklaces with 2n beads.
%C For n>=4 also number of Napier cycle types for dimension d=n-3. See Böhm link. - _Hugo Pfoertner_, Oct 01 2013
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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
%H Vincenzo Librandi, <a href="/A007147/b007147.txt">Table of n, a(n) for n = 1..1000</a>
%H Oswin Aichholzer and Anna Brötzner, <a href="https://doi.org/10.57717/cgt.v3i2.46">Bicolored Order Types</a>, Comp. Geom. Topology (2024) Vol. 3, No. 2, 3:1-3:17.
%H Amos Altshuler and Peter McMullen, <a href="http://dx.doi.org/10.1112/S0025579300004873">The number of simplicial neighbourly d-polytopes with d + 3 vertices</a>, Mathematika, 20(02):263-266, 1973., Theorem 1, p. 263.
%H Johannes Böhm, <a href="http://www.minet.uni-jena.de/Math-Net/reports/sources/2008/08-08report.pdf">Generalized hyperbolic Napier cycles and their hyperbolic kernels, Part III</a>, Jenaer Schriften zur Mathematik und Informatik, Math/inf/06/08, 2008.
%H Moritz Firsching, <a href="http://arxiv.org/abs/1508.02531">Realizability and inscribability for some simplicial spheres and matroid polytopes</a>, , arXiv:1508.02531 [math.MG], 2015. See Appendix A1.
%H Bernd Mulansky and Andreas Potschka, <a href="https://arxiv.org/abs/2404.01841">A zonogon approach for computing small polygons of maximum perimeter</a>, arXiv:2404.01841 [math.OC], 2024. See p. 9.
%H E. M. Palmer and R. W. Robinson, <a href="http://projecteuclid.org/euclid.pjm/1102711113">Enumeration of self-dual configurations</a>, Pacific J. Math., 110 (1984), 203-221.
%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F a(n) = (1/2) * (A016116(n-1) + A000016(n)).
%F a(n) = A263768(n) + 1. - _Bernd Mulansky_, Mar 08 2023
%t 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 *)
%o (PARI) a(n)= (1/2) *(2^((n-1)\2)+sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n))
%o (Sage)
%o def a(n):
%o 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)])
%o # _Moritz Firsching_, Aug 13 2015
%Y Cf. A000016, A016116, A263768.
%K nonn,nice,easy,changed
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_.
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