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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). 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A068346 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

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)