A063686 Triangular array: a(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

0,12

Comments

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

Links

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

Example

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;
...

Prog

(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(k<n-1, (k+1)*M[n-k-1, k+1] + sum(j=1, k, j*(M[n-j, k+1]-M[n-j, k])), if(k<n, n)))}
T(n)={my(M=S(R(n+1))); matid(n) + matrix(n, n, n, k, sumdiv(n, d, if(k<d, eulerphi(n/d)*M[d, k]))/n) }
my(M=T(10)); for(n=0, #M, for(k=0, n, print1(if(k==0, 1, M[n, k]), ", ")); print) \\ Andrew Howroyd, Oct 15 2017

Crossrefs

Cf. A032190, A048004, A048887.

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

Sequence in context: A049704 A047996 A227690 * A333159 A008327 A133687

Adjacent sequences: A063683 A063684 A063685 * A063687 A063688 A063689

Keyword

nonn,tabl

Author

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

Extensions

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

Status

approved