A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 1, 1, 1, 1, 4, 6, 4, 2, 1, 1, 1, 1, 7, 11, 8, 4, 2, 1, 1, 1, 1, 9, 19, 14, 8, 4, 2, 1, 1, 1, 1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1, 1, 18, 56, 50, 30, 16, 8, 4, 2, 1, 1, 1, 1, 30, 101, 96, 59, 32, 16, 8, 4, 2, 1, 1, 1

Offset: 0

Author

Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 22 2001

Keywords

• nonn
• tabl
• changed

Comments

Column k=1 appears to be A032190(n), n=2,3,...

Examples

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 2, 1, 1, 1;
  1, 2, 2, 1, 1, 1;
  1, 4, 4, 2, 1, 1, 1;
  1, 4, 6, 4, 2, 1, 1, 1;
  1, 7, 11, 8, 4, 2, 1, 1, 1;
  1, 9, 19, 14, 8, 4, 2, 1, 1, 1;
  1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274 (Rows n=0..49)

Crossrefs

Cf. A032190, A048004, A048887.

Cf. A000358, A093305, A280218 (necklaces avoiding 00, 000, 0000).

Programs

PARI

\\ here R(n) is A048887 transposed
R(n)={Mat(vector(n, k, Col((1-x)/(1-2*x+x^(k+1)) - 1 + O(x*x^n))))}
S(M)={matrix(#M-1, #M-1, n, k, if(kAndrew Howroyd, Oct 15 2017

Extensions

T(0,0)=1 from Andrew Howroyd, Oct 15 2017