OFFSET
1,2
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..500
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
N. J. A. Sloane, Maple code for this and related sequences
FORMULA
Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (5*S2(n/d + 4, 5) - 50*S2(n/d + 3, 5) + 175*S2(n/d + 2, 5) - 250*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==30 mod 60] * (4*S2(n/d+4,5) - 41*S2(n/d+3,5) + 149*S2(n/d+2,5) - 226*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==20 mod 60 | d==40 mod 60] * (4*S2(n/d + 4, 5) - 42*S2(n/d + 3, 5) + 156*S2(n/d + 2, 5) - 238*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d + 4, 5) - 33*S2(n/d + 3, 5) + 129*S2(n/d + 2, 5) - 210*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d mod 60 in {12,24,36,48}] * (4*S2(n/d + 4, 5) - 40*S2(n/d + 3, 5) + 140*S2(n/d + 2, 5) - 200*S2(n/d+1, 5) + 96*S2(n/d, 5)) + [d=10 mod 60 | d==50 mod 60] * (3*S2(n/d + 4, 5) - 33*S2(n/d + 3, 5) + 130*S2(n/d + 2, 5) - 214*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d mod 60 in {6,18,42,54}] * (3*S2(n/d + 4, 5) - 31*S2(n/d + 3, 5) + 114*S2(n/d + 2, 5) - 176*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {5,25,35,55}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 95*S2(n/d + 2, 5) - 165*S2(n/d + 1, 5) + 100*S2(n/d, 5)) + [d mod 60 in {4,8,16,28,32,44,52,56}] * (3*S2(n/d + 4, 5) - 32*S2(n/d + 3, 5) + 121*S2(n/d + 2, 5) - 188*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 94*S2(n/d + 2, 5) - 160*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {2,14,22,26,34,38,46,58}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 95*S2(n/d + 2, 5) - 164*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (S2[n/d + 4, 5) - 13*S2(n/d + 3, 5) + 60*S2(n/d + 2, 5) - 115*S2(n/d + 1, 5) + 76*S2(n/d, 5))), where S2(n,k) is the Stirling subset number, A008277.
G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * log(1-5x^d) + [d==30 mod 60] * (3*log(1-5x^d) + log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) + log(1-2x^d)) / 3 + [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) + 2*log(1-3x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in {12,24,36,48}] * 4*log(1-5x^d) / 5 + [d=10 mod 60 | d==50 mod 60] * (5*log(1-5x^d) + 4*log(1-2x^d) + 3*log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] * (5*log(1-5x^d) + 2*log(1-3x^d) + 4*log(1-2x^d) + 9*log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] * (7*log(1-5x^d) + 5*log(1-2x^d)) / 15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) + 10*log(1-3x^d) + 15*log(1-x^d)) / 40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) + 20*log(1-2x^d) + 15*log(1-x^d)) / 60 +[d mod 60 in{1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-5x^d) + 10*log(1-3x^d) + 20*log(1-2x^d) + 45*log(1-x^d)) / 120).
(End)
MATHEMATICA
Adn[d_, n_] := Module[{ c, t1, t2}, t2 = 0; For[c = 1, c <= d, c++, If[Mod[d, c] == 0 , t2 = t2 + (x^c/c)*(E^(c*z) - 1)]]; t1 = E^t2; t1 = Series[t1, {z, 0, n+1}]; Coefficient[t1, z, n]*n!]; Pn[n_] := Module[{ d, e, t1}, t1 = 0; For[d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*Adn[d, n/d]/n]]; t1/(1 - x)]; Pnq[n_, q_] := Module[{t1}, t1 = Series[Pn[n], {x, 0, q+1}] ; Coefficient[t1, x, q]]; a[n_] := Pnq[n, 5]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)
(* this Mathematica program uses Gilbert and Riordan's recurrence formula, which they recommend for calculations: *)
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[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]
/(n (1 - x)), {x, 0, 5}], {n, 1, 40}] (* Robert A. Russell, Feb 24 2018 *)
From Robert A. Russell, May 29 2018: (Start)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], 5 StirlingS2[n/#+4, 5] - 50 StirlingS2[n/#+3, 5] + 175 StirlingS2[n/#+2, 5] - 250 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 30], 4 StirlingS2[n/#+4, 5] - 41 StirlingS2[n/#+3, 5] + 149 StirlingS2[n/#+2, 5] - 226 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 20], 4 StirlingS2[n/#+4, 5] - 42 StirlingS2[n/#+3, 5] + 156 StirlingS2[n/#+2, 5] - 238 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 15], 3 StirlingS2[n/#+4, 5] - 33 StirlingS2[n/#+3, 5] + 129 StirlingS2[n/#+2, 5] - 210 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+4, 5] - 40 StirlingS2[n/#+3, 5] + 140 StirlingS2[n/#+2, 5] - 200 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 10], 3 StirlingS2[n/#+4, 5] - 33 StirlingS2[n/#+3, 5] + 130 StirlingS2[n/#+2, 5] - 214 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 6], 3 StirlingS2[n/#+4, 5] - 31 StirlingS2[n/#+3, 5] + 114 StirlingS2[n/#+2, 5] - 176 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 5], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 95 StirlingS2[n/#+2, 5] - 165 StirlingS2[n/#+1, 5] + 100 StirlingS2[n/#, 5], Divisible[#, 4], 3 StirlingS2[n/#+4, 5] - 32 StirlingS2[n/#+3, 5] + 121 StirlingS2[n/#+2, 5] - 188 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 3], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 94 StirlingS2[n/#+2, 5] - 160 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 2], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 95 StirlingS2[n/#+2, 5] - 164 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], True, StirlingS2[n/#+4, 5] - 13 StirlingS2[n/#+3, 5] + 60 StirlingS2[n/#+2, 5] - 115 StirlingS2[n/#+1, 5] + 76 StirlingS2[n/#, 5]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[1-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 60], Log[1-5x^d], Divisible[d, 30], (3 Log[1-5x^d] + Log[1-x^d]) / 4, Divisible[d, 20], (2 Log[1-5x^d] + Log[1-2x^d]) / 3, Divisible[d, 15], (3 Log[1-5x^d] + 2 Log[1-3x^d] + 3 Log[1-x^d]) / 8, Divisible[d, 12], 4 Log[1-5x^d] / 5, Divisible[d, 10], (5 Log[1-5x^d] + 4 Log[1-2x^d] + 3 Log[1-x^d]) / 12, Divisible[d, 6], (11 Log[1-5x^d] + 5 Log[1-x^d]) / 20, Divisible[d, 5], (5 Log[1-5x^d] + 2 Log[1-3x^d] + 4 Log[1-2x^d] + 9 Log[1-x^d]) / 24, Divisible[d, 4], (7 Log[1-5x^d] + 5 Log[1-2x^d]) / 15, Divisible[d, 3], (7 Log[1-5x^d] + 10 Log[1-3x^d] + 15 Log[1-x^d]) / 40, Divisible[d, 2], (13 Log[1-5x^d] + 20 Log[1-2x^d] + 15 Log[1-x^d]) / 60, True, (Log[1-5x^d] + 10 Log[1-3x^d] + 20 Log[1-2x^d] + 45 Log[1-x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved