A263657 Table T(m, n) of number of (0, 1)-necklaces without zigzags with m 1's and n 0's, read by antidiagonals (see reference for precise definition).

0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 3, 3, 3, 1, 1, 0, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 1, 0, 1, 1, 0, 1, 1, 4, 5, 7, 5, 4, 1, 1, 0, 1

Offset: 0

Author

Felix Fröhlich, Oct 23 2015

Keywords

• nonn
• tabl

Comments

See figure 2 on page 16 in the reference.

A zigzag is a substring which is either 010 or 101. The necklaces 01 and 10 are considered zigzags. Necklaces do not allow turnover.

Examples

			Table starts:
0 1 1 1 1  1  1  1   1   1   1    1    1    1    1    1 ...
1 0 0 0 0  0  0  0   0   0   0    0    0    0    0    0 ...
1 0 1 1 1  1  1  1   1   1   1    1    1    1    1    1 ...
1 0 1 1 1  1  1  1   1   1   1    1    1    1    1    1 ...
1 0 1 1 2  2  3  3   4   4   5    5    6    6    7    7 ...
1 0 1 1 2  3  4  5   6   7   8    9   10   11   12   13 ...
1 0 1 1 3  4  7  8  11  14  17   20   25   28   33   38 ...
1 0 1 1 3  5  8 12  17  23  30   38   47   57   68   80 ...
1 0 1 1 4  6 11 17  27  37  52   68   90  112  141  171 ...
1 0 1 1 4  7 14 23  37  57  82  115  157  207  268  341 ...
1 0 1 1 5  8 17 30  52  82 128  185  265  363  491  644 ...
1 0 1 1 5  9 20 38  68 115 185  285  423  608  850 1160 ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..819
• E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Crossrefs

Main diagonal is A263658. Antidiagonal sums are A263659.

Cf. A007039, A263655, A263656.

Extensions

a(45)-a(90) from Andrew Howroyd, Feb 26 2017