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

1,2

Comments

Turning over will not create a new bracelet.

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]

Formula

T(n, k) = Sum_{d|n} mu(n/d) * A081720(d,k) for k<=n. Corrected Jan 22 2022

Example

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 ...
  ...

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

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 *)

Crossrefs

Columns 2-6 are A001371, A032294, A032295, A032296, A056347.

Cf. A081720, A273891, A276543, A152176.

Sequence in context: A208544 A208535 A284856 * A294438 A074650 A284871

Adjacent sequences: A276547 A276548 A276549 * A276551 A276552 A276553

Keyword

nonn,tabl,easy

Author

Andrew Howroyd, Apr 09 2017

Status

approved