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 A056345 Number of bracelets of length n using exactly five different colored beads. 4
 0, 0, 0, 0, 12, 150, 1200, 7905, 46400, 255636, 1346700, 6901725, 34663020, 171786450, 843130688, 4110958530, 19951305240, 96528492700, 466073976900, 2247627076731, 10832193571460, 52194109216950 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Turning over will not create a new bracelet. 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..22. FORMULA a(n) = A032276(n) - 5*A032275(n) + 10*A027671(n) - 10*A000029(n) + 5. From Robert A. Russell, Sep 27 2018: (Start) a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=5 is the number of colors and S2 is the Stirling subset number A008277. G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=5 is the number of colors. (End) EXAMPLE For a(5)=12, pair up the 24 permutations of BCDE, each with its reverse, such as BCDE-EDCB. Precede the first of each pair with an A, such as ABCDE. These are the 12 arrangements, all chiral. If we precede the second of each pair with an A, such as AEDCB, we get the chiral partner of each. - Robert A. Russell, Sep 27 2018 MATHEMATICA t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}]; a[n_] := T[n, 5]; Array[a, 22] (* Jean-François Alcover, Nov 05 2017, after Andrew Howroyd *) k=5; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n, 1, 30}] (* Robert A. Russell, Sep 27 2018 *) CROSSREFS Column 5 of A273891. Equals (A056285 + A056491) / 2 = A056285 - A305544 = A305544 + A056491. Sequence in context: A154733 A305544 A056351 * A264233 A068768 A053507 Adjacent sequences: A056342 A056343 A056344 * A056346 A056347 A056348 KEYWORD nonn AUTHOR Marks R. Nester STATUS approved

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Last modified August 4 01:40 EDT 2024. Contains 374905 sequences. (Running on oeis4.)