A263655 - OEIS

%I #22 Jul 01 2018 15:27:27

%S 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,

%T 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,

%U 10,0,1,1,0,11,11,33,44,44,33,11,11,0,1

%N 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).

%C See page 5, figure 1 in the reference.

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

%H Andrew Howroyd, <a href="/A263655/b263655.txt">Table of n, a(n) for n = 0..819</a>

%H E. Munarini and N. Z. Salvi, <a href="http://www.emis.de/journals/INTEGERS/papers/d19/d19.Abstract.html">Circular Binary Strings without Zigzags</a>, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

%F From _Andrew Howroyd_, Feb 26 2017: (Start)

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

%F   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)).

%F 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.

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

%e Table starts:

%e 0 1  1  1  1   1   1   1   1    1    1    1    1 ...

%e 1 0  0  0  0   0   0   0   0    0    0    0    0 ...

%e 1 0  4  5  6   7   8   9  10   11   12   13   14 ...

%e 1 0  5  6  7   8   9  10  11   12   13   14   15 ...

%e 1 0  6  7 12  18  25  33  42   52   63   75   88 ...

%e 1 0  7  8 18  30  44  60  78   98  120  144  170 ...

%e 1 0  8  9 25  44  70 104 147  200  264  340  429 ...

%e 1 0  9 10 33  60 104 168 255  368  510  684  893 ...

%e 1 0 10 11 42  78 147 255 412  629  918 1292 1765 ...

%e 1 0 11 12 52  98 200 368 629 1014 1558 2300 3283 ...

%e 1 0 12 13 63 120 264 510 918 1558 2514 3885 5786 ...

%t max = 11;

%t U[r_, k_] := Binomial[r - k + 2*Floor[k/3], Floor[k/3]];

%t V[r_, k_] := Binomial[r - Ceiling[k/2] - 1, Floor[k/2]];

%t T[0, 0] = T[1, 1] = 0;

%t T[0, _] = T[_, 0] = 1;

%t 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 T[n_, m_] /; m > n := T[m, n];

%t Table[T[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)

%Y Main diagonal is A263656. Antidiagonal sums are A007039.

%Y Cf. A263657, A263658, A263659.

%K tabl,nonn

%O 0,13

%A _Felix Fröhlich_, Oct 23 2015

%E a(66)-a(77) from _Andrew Howroyd_, Feb 26 2017