A032239
Number of identity bracelets of n beads of 2 colors.
Original entry on oeis.org
2, 1, 0, 0, 0, 1, 2, 6, 14, 30, 62, 127, 252, 493, 968, 1860, 3600, 6902, 13286, 25446, 48914, 93775, 180314, 346420, 666996, 1284318, 2477328, 4781007, 9240012, 17870709, 34604066, 67058880, 130084990, 252545160, 490722342
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
- C. G. Bower, Transforms (2)
- Petros Hadjicostas, Formulas for chiral bracelets, 2019; see Section 5.
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
- Index entries for sequences related to bracelets
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m = 2; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d] (m^(n/d)/n - If[OddQ[n/d], m^((n/d + 1)/2), ((m + 1) m^(n/(2 d))/2)]), {d, Divisors[n]}]/2, {n, 3, 20}] (* Robert A. Russell, Mar 18 2013 *)
mx=40;gf[x_,k_]:=Sum[MoebiusMu[n]*(-Log[1-k*x^n]/n-Sum[Binomial[k,i]x^(n i),{i,0,2}]/(1-k x^(2n)))/2,{n,mx}];ReplacePart[Rest[CoefficientList[Series[gf[x,2],{x,0,mx}],x]],{1->2,2->1}] (* Herbert Kociemba, Nov 29 2016 *)
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a(n)={if(n<3, binomial(2,n), sumdiv(n, d, moebius(n/d)*(2^d/n - if(d%2, 2^((d+1)/2), 3*2^(d/2)/2)))/2)} \\ Andrew Howroyd, Sep 12 2019
A305541
Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 35, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 318, 2487, 7845, 11970, 8820, 2520, 0, 14, 934, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440, 0, 62, 7503, 158220, 1344900, 5873760, 14658840, 21772800, 19051200, 9072000, 1814400
Offset: 1
Triangle T(n,k) begins:
0;
0, 0;
0, 0, 1;
0, 0, 3, 3;
0, 0, 12, 24, 12;
0, 1, 35, 124, 150, 60;
0, 2, 111, 588, 1200, 1080, 360;
0, 6, 318, 2487, 7845, 11970, 8820, 2520;
0, 14, 934, 10240, 46280, 105840, 129360, 80640, 20160;
0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440;
...
For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.
For T(4,4)=3, the chiral pairs are ABCD-ADCB, ABDC-ACDB, and ACBD-ADBC.
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Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten
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T(n,k) = {-k!*(stirling((n+1)\2,k,2) + stirling(n\2+1,k,2))/4 + k!*sumdiv(n,d, eulerphi(d)*stirling(n/d,k,2))/(2*n)} \\ Andrew Howroyd, Sep 13 2019
A326660
Number of n-bead asymmetric bracelets with exactly 3 different colored beads.
Original entry on oeis.org
0, 0, 1, 3, 12, 34, 111, 315, 933, 2622, 7503, 21033, 59472, 167118, 472120, 1332945, 3777600, 10720869, 30516447, 87032994, 248820704, 712743768, 2045784183, 5882367570, 16942974048, 48876558318, 141204944529, 408494941773, 1183247473872, 3431450670601
Offset: 1
Case n = 4: There are 3 distinct asymmetric bracelets with exactly 3 colors which are aabc, abbc, abcc.
A326789
Number of n-bead asymmetric bracelets with exactly 4 different colored beads.
Original entry on oeis.org
0, 0, 0, 3, 24, 124, 588, 2484, 10240, 40464, 158220, 608951, 2333520, 8894616, 33864340, 128791140, 490027200, 1865614976, 7110959340, 27138170397, 103717719412, 396965536224, 1521562700988, 5840509149020, 22450188684264, 86412086034120, 333035003533660
Offset: 1
A326888
Number of length n asymmetric bracelets with integer entries that cover an initial interval of positive integers.
Original entry on oeis.org
0, 0, 1, 6, 48, 369, 3341, 33960, 393467, 5111052, 73753685, 1170468816, 20263758984, 380047813297, 7676106093000, 166114206886740, 3834434320842720, 94042629507381957, 2442147034668044933, 66942194905675830162, 1931543452344523775050, 58519191359155952404837
Offset: 1
For n = 4, the six asymmetric bracelets are 1123, 1223, 1233, 1234, 1243, 1324.
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\\ U(n,k) is A309528
U(n,k)={sumdiv(n, d, moebius(n/d)*(k^d/n - if(d%2, k^((d+1)/2), (k+1)*k^(d/2)/2)))/2}
a(n)={sum(k=1, n, U(n,k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}
Showing 1-5 of 5 results.
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