A056295 Number of n-bead necklace structures using exactly two different colored beads.

0, 1, 1, 3, 3, 7, 9, 19, 29, 55, 93, 179, 315, 595, 1095, 2067, 3855, 7315, 13797, 26271, 49939, 95419, 182361, 349715, 671091, 1290871, 2485533, 4794087, 9256395, 17896831, 34636833, 67110931, 130150587, 252648991, 490853415, 954444607, 1857283155, 3616828363

Offset

1,4

Comments

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.

Formula

a(n) = A000013(n) - 1.

From Robert A. Russell, Mar 08 2018: (Start)

G.f.: Sum_{ d>0 } phi(d)*(2*log(1-x^d) - (1+[d == 0 mod 2])*log(1-2*x^d)) / (2*d);

a(n) = (1/n)*Sum_{d|n} phi(d) * S2(n/d + [d == 0 mod 2], 2), where S2(n, k) is the Stirling subset number, A008277. (End)

Example

For a(7) = 9, the color patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AABAABB, AABABAB, AAABABB, and AAABBAB. The first seven are achiral; the last two are a chiral pair. - Robert A. Russell, Mar 08 2018

Maple

See A000013.

Mathematica

Table[DivisorSum[n, EulerPhi[#] If[OddQ[#], StirlingS2[n/#, 2], StirlingS2[n/#+1, 2]]&]/n, {n, 1, 30}] (* Robert A. Russell, Feb 20 2018 *)

Crossrefs

Column 2 of A152175.

Cf. A000013, A052823.

Sequence in context: A122012 A185306 A320314 * A117525 A075149 A161618

Adjacent sequences: A056292 A056293 A056294 * A056296 A056297 A056298

Keyword

nonn,easy

Author

Marks R. Nester

Status

approved