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

Author

Marks R. Nester

Keywords

• nonn

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

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

Crossrefs

Column 3 of A152175.

Cf. A000013, A002076, A056283.

Programs

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 *)

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)