A365094 Triangle read by rows: T(n,k) is the number of n-sided cycles with the property that one makes k turns to the right while following its edges.

1, 0, 0, 1, 1, 0, 4, 0, 1, 2, 5, 5, 5, 5, 2, 9, 12, 21, 36, 21, 12, 9, 31, 49, 147, 133, 133, 147, 49, 31, 128, 328, 652, 792, 1240, 792, 652, 328, 128, 708, 1719, 3717, 6735, 7281, 7281, 6735, 3717, 1719, 708, 4015, 10320, 28585, 43780, 58120, 73240, 58120, 43780, 28585, 10320, 4015

Offset: 3

Author

Ludovic Schwob, Aug 21 2023

Keywords

• nonn
• tabf

Comments

Cycles that differ by rotation or reflection are counted separately. By "n-sided cycles" we mean the cycles that can be drawn by connecting n equally spaced points on a circle (possibly self-intersecting).

Examples

			Triangle begins:
   1,   0,   0,   1;
   1,   0,   4,   0,   1;
   2,   5,   5,   5,   5,   2;
   9,  12,  21,  36,  21,  12,   9;
  31,  49, 147, 133, 133, 147,  49,  31;
		

Links

• Ludovic Schwob, Table of n, a(n) for n = 3..1481
• Ludovic Schwob, Illustration of T(6,k), 0<=k<=6

Crossrefs

Row sums give A000142(n-1) (number of cycles of length n).

Cf. A295264, A342968, A008292.

Formula

T(n,0) = T(n,n) = A295264(n).