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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

Table of n, a(n) for n=1..28.

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

Marks R. Nester

STATUS

approved

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Last modified April 4 21:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)