A276550 Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.
1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 7, 3, 0, 6, 10, 16, 15, 6, 0, 7, 15, 30, 45, 36, 8, 0, 8, 21, 50, 105, 132, 79, 16, 0, 9, 28, 77, 210, 372, 404, 195, 24, 0, 10, 36, 112, 378, 882, 1460, 1296, 477, 42, 0, 11, 45, 156, 630, 1848, 4220, 5890, 4380, 1209, 69, 0
Offset: 1
Examples
Table starts: 1 2 3 4 5 6 7 8 ... 0 1 3 6 10 15 21 28 ... 0 2 7 16 30 50 77 112 ... 0 3 15 45 105 210 378 630 ... 0 6 36 132 372 882 1848 3528 ... 0 8 79 404 1460 4220 10423 22904 ... 0 16 195 1296 5890 20640 60021 151840 ... 0 24 477 4380 25275 107100 364854 1057392 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
- G. Melançon, C. Reutenauer, On a Class of Lyndon Words Extending Christoffel Words and Related to a Multidimensional Continued Fraction Algorithm, J. Int. Seq. 16 (2013) #13.9.7, Corollary 6.
- 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]
Crossrefs
Programs
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Maple
A276550 := proc(n,k) local d ; add( numtheory[mobius](n/d)*A081720(d,k),d=numtheory[divisors](n)) ; end proc: seq(seq(A276550(n,d-n),n=1..d-1),d=2..10) ; # R. J. Mathar, Jan 22 2022
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Mathematica
t[n_, k_] := Sum[EulerPhi[d] k^(n/d), {d, Divisors[n]}]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4; T[n_, k_] := Sum[MoebiusMu[d] t[n/d, k], {d, Divisors[n]}]; Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 26 2020 *)
Formula
T(n, k) = Sum_{d|n} mu(n/d) * A081720(d,k) for k<=n. Corrected Jan 22 2022
Comments