A330341
Triangle read by rows: T(n,k) is the number of n-bead bracelets using exactly k colors with no adjacent beads having the same color.
Original entry on oeis.org
0, 0, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 3, 12, 12, 0, 1, 10, 46, 90, 60, 0, 0, 9, 120, 480, 720, 360, 0, 1, 27, 384, 2235, 5670, 6300, 2520, 0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160, 0, 1, 75, 2904, 38484, 217152, 604800, 876960, 635040, 181440
Offset: 1
Triangle begins:
0;
0, 1;
0, 0, 1;
0, 1, 3, 3;
0, 0, 3, 12, 12;
0, 1, 10, 46, 90, 60;
0, 0, 9, 120, 480, 720, 360;
0, 1, 27, 384, 2235, 5670, 6300, 2520;
0, 0, 29, 980, 9380, 36960, 68880, 60480, 20160;
...
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\\ here U(n, k) is A208544(n, k) for n > 1.
U(n, k) = (sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n + if(n%2, 1-k, k*(k-1)^(n/2)/2))/2;
T(n, k)={sum(j=1, k, (-1)^(k-j)*binomial(k, j)*U(n, j))}
A093367
Number of n-bead necklaces using exactly three colors with no adjacent beads having the same color.
Original entry on oeis.org
0, 0, 2, 3, 6, 11, 18, 33, 58, 105, 186, 349, 630, 1179, 2190, 4113, 7710, 14599, 27594, 52485, 99878, 190743, 364722, 699249, 1342182, 2581425, 4971066, 9587577, 18512790, 35792565, 69273666, 134219793, 260301174, 505294125, 981706830, 1908881897, 3714566310
Offset: 1
a(3) = 2 because the two necklaces 123 and 132 have no adjacent equal elements. - _Andrew Howroyd_, Dec 21 2019
- David W. Hobill and Scott MacDonald (zeened(AT)shaw.ca), Preprint, 2004.
- P. K.-H. Ma and J. Wainwright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Relativity Today, 1994.
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Table[Mod[n, 2] - 3 + DivisorSum[n, EulerPhi[n/#] 2^# &]/n, {n, 37}] (* Michael De Vlieger, Dec 22 2019 *)
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a(n)={n%2 - 3 + sumdiv(n, d, eulerphi(n/d)*2^d)/n} \\ Andrew Howroyd, Dec 21 2019
a(1)-a(2) prepended and terms a(20) and beyond from
Andrew Howroyd, Dec 21 2019
A330620
Number of length n necklaces with entries covering an initial interval of positive integers and no adjacent entries equal.
Original entry on oeis.org
0, 1, 2, 10, 54, 392, 3378, 34120, 393738, 5112406, 73756026, 1170482186, 20263782630, 380047964920, 7676106365966, 166114208828980, 3834434324386350, 94042629535109500, 2442147034719168714, 66942194906112161302, 1931543452345335094678, 58519191359163454708564
Offset: 1
Case n=4: there are the following 10 necklaces:
1212,
1213, 1232, 1323,
1234, 1243, 1324, 1342, 1423, 1432.
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\\ here U(n, k) is A208535(n, k) for n > 1.
U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)}
a(n)={if(n<1, n==0, sum(j=1, n, U(n,j)*sum(k=j, n, (-1)^(k-j)*binomial(k, j))))}
Showing 1-3 of 3 results.
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