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A056295
Number of n-bead necklace structures using exactly two different colored beads.
9
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
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.
Sequence in context: A122012 A185306 A320314 * A117525 A075149 A161618
KEYWORD
nonn,easy
STATUS
approved