2,13

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.

A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0

to floor(n/2) beads 0, i.e. 0 <= k <= floor(n/2). If n is even, the bracelet

0101...01 with n/2 beads of each kind, does not has 00 and we cannot change

any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet

0101...011 with (n-1)/2 beads 0.

The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0

with 00 prohibited, is equal to the number of binary bracelets with n-k beads,

k of them 0. See below.

Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1

(circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.

If we do this insertion k times, each time after a distinct 0 of B, we obtain a

bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.

On the contrary, Let B be a binary bracelet with n beads, k of them 0, with

00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet

of n-1 beads, k of them 0. (If not and we undo the removal, the configuration

obtained cannot be a bracelet and this is absurd.) If we repeat this removal k

times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

Array begins

1 1

1 1

1 1 1

1 1 1

1 1 2 1

1 1 2 1

1 1 3 2 1

1 1 3 3 1

1 1 4 4 3 1

...

A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +

A032279(5) = 1+1+4+4+3+1 = 14.

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514,

A032281, A005515, A032282, A005516. Row sums of array give A129526.

Sequence in context: A090677 A161097 A105240 this_sequence A161096 A165983 A083894

Adjacent sequences: A143651 A143652 A143653 this_sequence A143655 A143656 A143657

nonn,tabf

Washington Bomfim (webonfim(AT)bol.com.br), Aug 28 2008