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A343257 Triangle read by rows: T(n,k) is the number of n+2-sided polygons whose points lie on a circle and with the property that one makes k turns on itself, always in the same direction (right or left) while following its edges, 1 <= k <= ceiling(n/2).

%I A343257 #24 Jun 16 2021 02:08:23
%S A343257 1,1,1,1,1,8,1,29,1,1,80,47,1,193,513,1,1,432,3338,244,1,925,16633,
%T A343257 7305,1,1,1928,70713,103616,1186,1,3953,271441,990289,92145,1,1,8024,
%U A343257 972548,7438204,2717321,5536,1,16189,3321009,47629761,47637225,1076409,1
%N A343257 Triangle read by rows: T(n,k) is the number of n+2-sided polygons whose points lie on a circle and with the property that one makes k turns on itself, always in the same direction (right or left) while following its edges, 1 <= k <= ceiling(n/2).
%C A343257 Polygons that differ by rotation or reflection are counted separately.
%C A343257 The polygons considered here are those that can be drawn by connecting n+2 equally spaced points on a circle (possibly self-intersecting).
%C A343257 The number of turns a polygon makes on itself while following its edges is called the turning number. See the Wikipedia article for additional explanation. The condition that the turns are in the same direction means that all the internal angles are less than 180 degrees (stars are allowed, but figure of eights are not).
%H A343257 Ludovic Schwob, <a href="/A343257/a343257.pdf">Illustration of T(6,k), 1<=k<=3</a>
%H A343257 Wikipedia, <a href="https://en.wikipedia.org/wiki/Winding_number#Turning_number">Turning number</a>
%F A343257 T(n,1)=1 and T(2*n-1,n)=1 for all n>=1: the only solutions are the polygons with respective Schläfli symbols {n+2} and {2*n+1/n}.
%F A343257 T(n,2) = A030110(n+1) for all n>=1.
%F A343257 T(2*n,n-1) = A029760(n-2) for all n>=2.
%e A343257 Triangle begins:
%e A343257      1;
%e A343257      1;
%e A343257      1,    1;
%e A343257      1,    8;
%e A343257      1,   29,    1;
%e A343257      1,   80,   47;
%e A343257      1,  193,  513,   1;
%o A343257 (PARI)
%o A343257 B(n,m,x)={
%o A343257   local(Cache=Map());
%o A343257   my(recurse(k,p,q,b) = my(hk=[k,p,q,b], z); if(!mapisdefined(Cache, hk, &z),
%o A343257   z = if(k==0, q>p && q>m, sum(j=1, n-(q-p)%n, my(r=(q+j)%n); if(!bittest(b,r), if(r<q, x, 1)*self()(k-1, q, r, b+(1<<r)) )));
%o A343257   mapput(Cache, hk, z)); z);
%o A343257   recurse(n-2, 0, m, 1+(1<<m));
%o A343257 }
%o A343257 T(n)={Vecrev(sum(i=1, n-1, B(n, i,'x)))}
%o A343257 { for(n=3, 12, print(T(n))) } \\ _Andrew Howroyd_, May 15 2021
%Y A343257 Row sums give A295264(n+1).
%Y A343257 Cf. A029760, A030110, A342968.
%K A343257 nonn,tabf
%O A343257 1,6
%A A343257 _Ludovic Schwob_, Apr 09 2021
%E A343257 a(31)-a(49) from _Andrew Howroyd_, May 15 2021