A056296 Number of n-bead necklace structures using exactly three different colored beads.

0, 0, 1, 2, 5, 18, 43, 126, 339, 946, 2591, 7254, 20125, 56450, 158355, 446618, 1262225, 3580686, 10181479, 29032254, 82968843, 237645250, 682014587, 1960981598, 5647919645, 16292761730, 47069104613, 136166703562, 394418199725, 1143822046786, 3320790074371

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

Table of n, a(n) for n=1..31.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Formula

a(n) = A002076(n) - A000013(n).

From Robert A. Russell, May 29 2018: (Start)

a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 6] * (S2(n/d + 2, 3) - S2(n/d + 1, 3)) + [d==3 mod 6] * (S2(n/d + 2, 3) - 3*S2(n/d + 1, 3) + 3*S2(n/d, 3)) + [d==2 mod 6 | d==4 mod 6] * (2*S2(n/d + 1, 3) - 2*S2(n/d, 3)) + [d==1 mod 6 | d=5 mod 6] * S2(n/d, 3)), where S2(n,k) is the Stirling subset number, A008277.

G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 6] * (log(1-3x^d) - log(1-x^d)) + [d==3 mod 6] * (log(1-3x^d) - log(1-2x^d) + log(1-x^d)) / 2 + [d==2 mod 6 | d==4 mod 6] * (2*log(1-3x^d) - 3*log(1-2x^d)) / 3 + [d==1 mod 6 | d=5 mod 6] * (log(1-3x^d) - 3*log(1-2x^d) + 3*log(1-x^d)) / 6).

(End)

Mathematica

Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &], Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 3], {n, 1, 40}] (* Robert A. Russell, Feb 23 2018 *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 6], StirlingS2[n/#+2, 3] - StirlingS2[n/#+1, 3], Divisible[#, 3], StirlingS2[n/#+2, 3] - 3 StirlingS2[n/#+1, 3] + 3 StirlingS2[n/#, 3], Divisible[#, 2], 2 StirlingS2[n/#+1, 3] - 2 StirlingS2[n/#, 3], True, StirlingS2[n/#, 3]] &], {n, 1, 40}] (* Robert A. Russell, May 29 2018*)
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 6], Log[1 - 3x^d] - Log[1 - 2x^d], Divisible[d, 3] , (Log[1 - 3x^d] - Log[1 - 2x^d] + Log[1 - x^d]) / 2, Divisible[d, 2], (2 Log[1 - 3x^d] - 3 Log[1 - 2x^d]) / 3, True, (Log[1 - 3x^d] - 3Log[1 - 2x^d] + 3 Log[1 - x^d]) / 6], {d, 1, mx}], {x, 0, mx}], x], 1] (* Robert A. Russell, May 29 2018 *)

Crossrefs

Column 3 of A152175.

Cf. A000013, A002076, A056283.

Sequence in context: A048221 A183365 A295905 * A293964 A275079 A148417

Adjacent sequences: A056293 A056294 A056295 * A056297 A056298 A056299

Keyword

nonn

Author

Marks R. Nester

Status

approved