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A056357 Number of bracelet structures using exactly two different colored beads.

%I A056357 #26 Oct 24 2019 23:02:09
%S A056357 0,1,1,3,3,7,8,17,22,43,62,121,189,361,611,1161,2055,3913,7154,13647,
%T A056357 25481,48733,92204,176905,337593,649531,1246862,2405235,4636389,
%U A056357 8964799,17334800,33588233,65108061,126390031,245492243,477353375,928772649,1808676325
%N A056357 Number of bracelet structures using exactly two different colored beads.
%C A056357 Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.
%C A056357 Also the number of distinct twills of period n. [Grünbaum and Shephard]
%D A056357 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A056357 Andrew Howroyd, <a href="/A056357/b056357.txt">Table of n, a(n) for n = 1..1000</a>
%H A056357 B. Grünbaum and G. C. Shephard, <a href="http://www.jstor.org/stable/2690105">Satins and twills: an introduction to the geometry of fabrics</a>, Math. Mag., 53 (1980), 139-161. See Theorem 2. [From _N. J. A. Sloane_, Jul 13 2011]
%F A056357 a(n) = A000011(n) - 1.
%F A056357 For an explicit formula see the Maple program.
%p A056357 with(numtheory);
%p A056357 rho:=n->(3+(-1)^n)/2;
%p A056357 f:=n->2^((n+rho(n))/2-2) + (1/(4*n))*(add(phi(d)*rho(d)*2^(n/d), d in divisors(n))) - 1;
%p A056357 # _N. J. A. Sloane_, Jul 13 2011
%o A056357 (PARI) a(n) = {if(n<1, 0, 2^(n\2-1) - 1 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; \\ _Andrew Howroyd_, Oct 24 2019
%Y A056357 Column 2 of A152176.
%Y A056357 Cf. A056295.
%K A056357 nonn
%O A056357 1,4
%A A056357 _Marks R. Nester_
%E A056357 Terms a(32) and beyond from _Andrew Howroyd_, Oct 24 2019