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 A056292 Number of n-bead necklace structures using a maximum of four different colored beads. 11
 1, 2, 3, 7, 11, 39, 103, 367, 1235, 4439, 15935, 58509, 215251, 799697, 2983217, 11187567, 42109451, 159082753, 602809327, 2290684251, 8726308317, 33318661277, 127479700199, 488672302909, 1876500180291, 7217308815887, 27799998949873, 107228568948547 (list; graph; refs; listen; history; text; internal format)
 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 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 12] * (4*S2(n/d+3, 4) - 24*S2(n/d+2, 4) + 44*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==6 mod 12] * (3*S2(n/d+3, 4) - 18*S2(n/d+2, 4) + 33*S2(n/d+1, 4) - 18*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (3*S2(n/d+3, 4) - 19*S2(n/d+2, 4) + 38*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 26*S2(n/d+1, 4) - 15*S2(n/d, 4)) + [d==2 mod 12 | d=10 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 27*S2[n/d+1,4) - 18*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * (S2(n/d+3, 4) - 8*S2(n/d+2, 4) + 20*S2(n/d+1, 4) - 15*S2(n/d, 4))), where S2(n, k) is the Stirling subset number, A008277. G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * log(1-4x^d) + [d==6 mod 12] * 3*log(1-4x^d) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d=9 mod 12] * (3*log(1-4x^d) + 2*log(1-2x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) + 6*log(1-2x^d) + 8*log(1-x^d)) / 24). (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, 4]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *) (* This 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, 4}], {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[#, 12], 4 StirlingS2[n/#+3, 4] - 24 StirlingS2[n/#+2, 4] + 44 StirlingS2[n/#+1, 4] - 24 StirlingS2[n/#, 4], Divisible[#, 6], 3 StirlingS2[n/#+3, 4] - 18 StirlingS2[n/#+2, 4] + 33 StirlingS2[n/#+1, 4] - 18 StirlingS2[n/#, 4], Divisible[#, 4], 3 StirlingS2[n/#+3, 4] - 19 StirlingS2[n/#+2, 4] + 38 StirlingS2[n/#+1, 4] - 24 StirlingS2[n/#, 4], Divisible[#, 3], 2 StirlingS2[n/#+3, 4] - 13 StirlingS2[n/#+2, 4] + 26 StirlingS2[n/#+1, 4] - 15 StirlingS2[n/#, 4], Divisible[#, 2], 2 StirlingS2[n/#+3, 4] - 13 StirlingS2[n/#+2, 4] + 27 StirlingS2[n/#+1, 4] - 18 StirlingS2[n/#, 4], True, StirlingS2[n/#+3, 4] - 8 StirlingS2[n/#+2, 4] + 20 StirlingS2[n/#+1, 4] - 15 StirlingS2[n/#, 4]] &], {n, 1, 40}] mx = 40; Drop[CoefficientList[Series[1 - Sum[(EulerPhi[d] / d) Which[   Divisible[d, 12], Log[1 - 4x^d], Divisible[d, 6],   3 Log[1 - 4x^d] / 4, Divisible[d, 4] ,   (2 Log[1 - 4x^d] + Log[1 - x^d]) / 3, Divisible[d, 3],   (3 Log[1 - 4x^d] + 2 Log[1 - 2x^d]) / 8,   Divisible[d, 2], (5 Log[1 - 4x^d] + 4 Log[1 - x^d]) / 12,   True, (Log[1 - 4x^d] + 6 Log[1 - 2x^d] + 8 Log[1 - x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1] (End) CROSSREFS Cf. A000013, A001868. Sequence in context: A338320 A056354 A072534 * A106125 A175171 A073609 Adjacent sequences:  A056289 A056290 A056291 * A056293 A056294 A056295 KEYWORD nonn AUTHOR STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)