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

1,4

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]

Formula

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

For an explicit formula see the Maple program.

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

Prog

(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

Crossrefs

Column 2 of A152176.

Cf. A056295.

Sequence in context: A108046 A286110 A116157 * A288728 A213942 A144554

Adjacent sequences: A056354 A056355 A056356 * A056358 A056359 A056360

Keyword

nonn

Author

Marks R. Nester

Extensions

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

Status

approved