A263655 Table T(m, n) of number of circular binary strings with m ones and n zeros without zigzags, read by antidiagonals (see reference for precise definition).

0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 4, 0, 1, 1, 0, 5, 5, 0, 1, 1, 0, 6, 6, 6, 0, 1, 1, 0, 7, 7, 7, 7, 0, 1, 1, 0, 8, 8, 12, 8, 8, 0, 1, 1, 0, 9, 9, 18, 18, 9, 9, 0, 1, 1, 0, 10, 10, 25, 30, 25, 10, 10, 0, 1, 1, 0, 11, 11, 33, 44, 44, 33, 11, 11, 0, 1

Offset

0,13

Comments

See page 5, figure 1 in the reference.

A zigzag is a substring which is either 010 or 101.

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.

Formula

From Andrew Howroyd, Feb 26 2017: (Start)

T(n,m) = Sum_{k>=0} U(m,k)*U(n,k) - 2*V(m,k)*V(n,k)*(-1)^k

where U(r,k)=binomial(r-k+2*floor(k/3), floor(k/3)), V(r,k)=binomial(r-ceiling(k/2)-1, floor(k/2)).

T(n,0)=1 for n>=1, T(n,1)=0 for n>=1, T(n,2)=n+2 for n>=2, T(n,3)=n+3 for n>=2.

T(n,4)=(n-1)*(n+4)/2 for n>=3, T(n,5)=(n-2)*(n+5) for n>=3. (End)

Example

Table starts:
0 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 ...
1 0  4  5  6   7   8   9  10   11   12   13   14 ...
1 0  5  6  7   8   9  10  11   12   13   14   15 ...
1 0  6  7 12  18  25  33  42   52   63   75   88 ...
1 0  7  8 18  30  44  60  78   98  120  144  170 ...
1 0  8  9 25  44  70 104 147  200  264  340  429 ...
1 0  9 10 33  60 104 168 255  368  510  684  893 ...
1 0 10 11 42  78 147 255 412  629  918 1292 1765 ...
1 0 11 12 52  98 200 368 629 1014 1558 2300 3283 ...
1 0 12 13 63 120 264 510 918 1558 2514 3885 5786 ...

Mathematica

max = 11;
U[r_, k_] := Binomial[r - k + 2*Floor[k/3], Floor[k/3]];
V[r_, k_] := Binomial[r - Ceiling[k/2] - 1, Floor[k/2]];
T[0, 0] = T[1, 1] = 0;
T[0, _] = T[_, 0] = 1;
T[n_ /; n >= 2, m_] /; m <= n := T[n, m] = Switch[m, 1, 0, 2, n + 2, 3, n + 3, _, Sum[ U[m, k]*U[n, k] - 2*V[m, k]*V[n, k]*(-1)^k, {k, 0, max-3}]];
T[n_, m_] /; m > n := T[m, n];
Table[T[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 01 2018, after Andrew Howroyd *)

Crossrefs

Main diagonal is A263656. Antidiagonal sums are A007039.

Cf. A263657, A263658, A263659.

Sequence in context: A348304 A006838 A061309 * A329078 A059064 A321316

Adjacent sequences: A263652 A263653 A263654 * A263656 A263657 A263658

Keyword

tabl,nonn

Author

Felix Fröhlich, Oct 23 2015

Extensions

a(66)-a(77) from Andrew Howroyd, Feb 26 2017

Status

approved