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

0,4

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).

Links

Table of n, a(n) for n=0..40.

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

Formula

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

Example

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

Crossrefs

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

Sequence in context: A302595 A113345 A078123 * A323312 A231774 A209170

Adjacent sequences: A342965 A342966 A342967 * A342969 A342970 A342971

Keyword

nonn,tabf,more

Author

Ludovic Schwob, Apr 01 2021

Status

approved