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
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
FORMULA
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
KEYWORD
nonn
AUTHOR
STATUS
approved