%I #10 Sep 16 2018 02:03:19
%S 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,
%T 1,1,4,5,4,1,1,1,5,7,8,3,1,1,1,5,8,10,5,1,1,1,6,10,16,10,4,1,1,1,6,12,
%U 20,16,7,1,1,1,7,14,29,26,16,4,1,1,1,7,16,35,38,26,8,1,1,1,8,19,47,57,50
%N Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).
%C The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
%C 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 have 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.
%C 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.
%C 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.
%C 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.
%C 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.
%e Array begins
%e 1 1
%e 1 1
%e 1 1 1
%e 1 1 1
%e 1 1 2 1
%e 1 1 2 1
%e 1 1 3 2 1
%e 1 1 3 3 1
%e 1 1 4 4 3 1
%e ...
%e A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
%e A032279(5) = 1+1+4+4+3+1 = 14.
%Y Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.
%K nonn,tabf
%O 2,13
%A _Washington Bomfim_, Aug 28 2008