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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A342972

KEYWORD

nonn,tabf,more

AUTHOR

Ludovic Schwob, Apr 01 2021

STATUS

approved

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Last modified July 29 02:36 EDT 2021. Contains 346340 sequences. (Running on oeis4.)