A342968 - OEIS

%I #25 Jun 13 2021 07:19:00

%S 1,0,1,2,1,5,6,1,24,28,8,119,183,57,1,832,1209,432,47,6255,9514,3760,

%T 630,1,54380,82636,36352,7828,244,515284,812714,383648,94997,7756,1,

%U 5454624,8727684,4377888,1243482,153536,1186

%N Irregular triangle read by rows: T(n,k) is the number of n+2-sided polygons with the property that one makes k turns on itself while following its edges.

%C Polygons that differ by rotation or reflection are counted separately.

%C By "n+2-sided polygons" we mean the polygons that can be drawn by connecting n+2 equally spaced points on a circle (possibly self-intersecting).

%C T(0,0)=1 by convention.

%C To compute the number of turns, follow the edges of the polygon, and add the angles of rotation: positive if turning left, negative if turning right. Then take the absolute value of the sum (see illustration).

%H Ludovic Schwob, <a href="/A342968/a342968.pdf">Illustration of T(5,k), 0 <= k <= 3</a>

%F T(2*n-1,n)=1 for all n >= 1: the only solution is the polygon with Schläfli symbol {2*n+1/n}.

%e Triangle begins:

%e      1;

%e      0,    1;

%e      2,    1;

%e      5,    6,    1;

%e     24,   28,    8;

%e    119,  183,   57,   1;

%Y Row sums give A001710(n+1) (number of polygons with n+2 sides).

%K nonn,tabf,more

%O 0,4

%A _Ludovic Schwob_, Apr 01 2021