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

0, 1, 1, 3, 3, 7, 8, 17, 22, 43, 62, 121, 189, 361, 611, 1161, 2055, 3913, 7154, 13647, 25481, 48733, 92204, 176905, 337593, 649531, 1246862, 2405235, 4636389, 8964799, 17334800, 33588233, 65108061, 126390031, 245492243, 477353375, 928772649, 1808676325

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure.

Also the number of distinct twills of period n. [Grünbaum and Shephard]

References

• 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]

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• B. Grünbaum and G. C. Shephard, Satins and twills: an introduction to the geometry of fabrics, Math. Mag., 53 (1980), 139-161. See Theorem 2. [From _N. J. A. Sloane_, Jul 13 2011]

Crossrefs

Column 2 of A152176.

Cf. A056295.

Programs

Maple

with(numtheory);
rho:=n->(3+(-1)^n)/2;
f:=n->2^((n+rho(n))/2-2) + (1/(4*n))*(add(phi(d)*rho(d)*2^(n/d), d in divisors(n))) - 1;
# N. J. A. Sloane, Jul 13 2011

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

Formula

a(n) = A000011(n) - 1.

For an explicit formula see the Maple program.

Extensions

Terms a(32) and beyond from Andrew Howroyd, Oct 24 2019