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 A056297 Number of n-bead necklace structures using exactly four different colored beads. 7
 0, 0, 0, 1, 2, 13, 50, 221, 866, 3437, 13250, 51075, 194810, 742651, 2823766, 10738881, 40843370, 155494751, 592614050, 2261625725, 8643289534, 33080920607, 126797503250, 486710971595, 1870851589554, 7201014763285, 27752927359726, 107092397450897 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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 a(n) = A056292(n) - A002076(n). From Robert A. Russell, May 29 2018: (Start) a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 12] * (S2(n/d + 3, 4) - 3*S2(n/d+2,4) + 2*S2(n/d + 1, 4)) + [d==6 mod 12] * (3*S2(n/d + 2, 4) - 9*S2(n/d + 1, 4) + 3*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (S2(n/d + 3, 4) - 5*S2(n/d + 2, 4) - 10*S2(n/d + 1, 4) - 8*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d + 2, 4) - 8*S2(n/d + 1, 4) - 2*S2(n/d,4)) + [d==2 mod 12 | d==10 mod 12] * (S2(n/d + 2, 4) - S2(n/d + 1, 4) + 9*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * S2(n/d, 4)), where S2(n,k) is the Stirling subset number, A008277. G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * (log(1-4x^d) - log(1-3x^d)) +[d==6 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d)) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) - 2*log(1-3x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d==9 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d) + 2*log(1-2x^d) - 4*log(1-x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) - 8*log(1-3x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) - 4*log(1-3x^d) + 6*log(1-2x^d) - 4*log(1-x^d)) / 24). (End) MATHEMATICA From Robert A. Russell, May 29 2018: (Start) 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, 4], {n, 1, 40}] (* after Gilbert and Riordan *) Table[(1/n) DivisorSum[n, EulerPhi[#] Which[ Divisible[#, 12], StirlingS2[n/#+3, 4] - 3 StirlingS2[n/#+2, 4] + 2 StirlingS2[n/#+1, 4], Divisible[#, 6], 3 StirlingS2[n/#+2, 4] - 9 StirlingS2[n/#+1, 4] + 6 StirlingS2[n/#, 4], Divisible[#, 4], StirlingS2[n/#+3, 4] - 5 StirlingS2[n/#+2, 4] + 10 StirlingS2[n/#+1, 4] - 8 StirlingS2[n/#, 4], Divisible[#, 3], 2 StirlingS2[n/#+2, 4] - 8 StirlingS2[n/#+1, 4] + 9 StirlingS2[n/#, 4], Divisible[#, 2], StirlingS2[n/#+2, 4] - StirlingS2[n/#+1, 4] - 2 StirlingS2[n/#, 4], True, StirlingS2[n/#, 4]] &], {n, 1, 40}] mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 12], Log[1-4x^d] - Log[1-3x^d], Divisible[d, 6], (3 Log[1-4x^d] - 4 Log[1-3x^d]) / 4, Divisible[d, 4], (2 Log[1-4x^d] - 2 Log[1-3x^d] + Log[1-x^d]) / 3, Divisible[d, 3], (3 Log[1-4x^d] - 4 Log[1-3x^d] + 2 Log[1-2x^d] - 4 Log[1-x^d]) / 8, Divisible[d, 2], (5 Log[1-4x^d] - 8 Log[1-3x^d] + 4 Log[1-x^d]) / 12, True, (Log[1-4x^d] - 4 Log[1-3x^d] + 6 Log[1-2x^d] - 4 Log[1-x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1] (End) CROSSREFS Column 4 of A152175. Cf. A002076, A056284, A056292. Sequence in context: A319759 A254784 A056305 * A241892 A037383 A034476 Adjacent sequences:  A056294 A056295 A056296 * A056298 A056299 A056300 KEYWORD nonn AUTHOR STATUS approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)