This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263655 #22 Jul 01 2018 15:27:27
%S A263655 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 A263655 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 A263655 10,0,1,1,0,11,11,33,44,44,33,11,11,0,1
%N 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).
%C A263655 See page 5, figure 1 in the reference.
%C A263655 A zigzag is a substring which is either 010 or 101.
%H A263655 Andrew Howroyd, <a href="/A263655/b263655.txt">Table of n, a(n) for n = 0..819</a>
%H A263655 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 A263655 From _Andrew Howroyd_, Feb 26 2017: (Start)
%F A263655 T(n,m) = Sum_{k>=0} U(m,k)*U(n,k) - 2*V(m,k)*V(n,k)*(-1)^k
%F A263655 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 A263655 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 A263655 T(n,4)=(n-1)*(n+4)/2 for n>=3, T(n,5)=(n-2)*(n+5) for n>=3. (End)
%e A263655 Table starts:
%e A263655 0 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e A263655 1 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A263655 1 0 4 5 6 7 8 9 10 11 12 13 14 ...
%e A263655 1 0 5 6 7 8 9 10 11 12 13 14 15 ...
%e A263655 1 0 6 7 12 18 25 33 42 52 63 75 88 ...
%e A263655 1 0 7 8 18 30 44 60 78 98 120 144 170 ...
%e A263655 1 0 8 9 25 44 70 104 147 200 264 340 429 ...
%e A263655 1 0 9 10 33 60 104 168 255 368 510 684 893 ...
%e A263655 1 0 10 11 42 78 147 255 412 629 918 1292 1765 ...
%e A263655 1 0 11 12 52 98 200 368 629 1014 1558 2300 3283 ...
%e A263655 1 0 12 13 63 120 264 510 918 1558 2514 3885 5786 ...
%t A263655 max = 11;
%t A263655 U[r_, k_] := Binomial[r - k + 2*Floor[k/3], Floor[k/3]];
%t A263655 V[r_, k_] := Binomial[r - Ceiling[k/2] - 1, Floor[k/2]];
%t A263655 T[0, 0] = T[1, 1] = 0;
%t A263655 T[0, _] = T[_, 0] = 1;
%t A263655 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 A263655 T[n_, m_] /; m > n := T[m, n];
%t A263655 Table[T[n - k, k], {n, 0, max}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)
%Y A263655 Main diagonal is A263656. Antidiagonal sums are A007039.
%Y A263655 Cf. A263657, A263658, A263659.
%K A263655 tabl,nonn
%O A263655 0,13
%A A263655 _Felix Fröhlich_, Oct 23 2015
%E A263655 a(66)-a(77) from _Andrew Howroyd_, Feb 26 2017