A342968 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.

1, 0, 1, 2, 1, 5, 6, 1, 24, 28, 8, 119, 183, 57, 1, 832, 1209, 432, 47, 6255, 9514, 3760, 630, 1, 54380, 82636, 36352, 7828, 244, 515284, 812714, 383648, 94997, 7756, 1, 5454624, 8727684, 4377888, 1243482, 153536, 1186

Offset: 0

Author

Ludovic Schwob, Apr 01 2021

Keywords

• nonn
• tabf
• more

Comments

Polygons that differ by rotation or reflection are counted separately.

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).

T(0,0)=1 by convention.

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).

Examples

			Triangle begins:
     1;
     0,    1;
     2,    1;
     5,    6,    1;
    24,   28,    8;
   119,  183,   57,   1;
		

Links

• Ludovic Schwob, Illustration of T(5,k), 0 <= k <= 3

Crossrefs

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

Formula

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