A295264 -id:A295264 - cp's OEIS frontend

A343257 Triangle read by rows: T(n,k) is the number of n+2-sided polygons whose points lie on a circle and with the property that one makes k turns on itself, always in the same direction (right or left) while following its edges, 1 <= k <= ceiling(n/2).

1, 1, 1, 1, 1, 8, 1, 29, 1, 1, 80, 47, 1, 193, 513, 1, 1, 432, 3338, 244, 1, 925, 16633, 7305, 1, 1, 1928, 70713, 103616, 1186, 1, 3953, 271441, 990289, 92145, 1, 1, 8024, 972548, 7438204, 2717321, 5536, 1, 16189, 3321009, 47629761, 47637225, 1076409, 1

Offset: 1

Ludovic Schwob, Apr 09 2021

Polygons that differ by rotation or reflection are counted separately.
The polygons considered here are those that can be drawn by connecting n+2 equally spaced points on a circle (possibly self-intersecting).
The number of turns a polygon makes on itself while following its edges is called the turning number. See the Wikipedia article for additional explanation. The condition that the turns are in the same direction means that all the internal angles are less than 180 degrees (stars are allowed, but figure of eights are not).

			Triangle begins:
     1;
     1;
     1,    1;
     1,    8;
     1,   29,    1;
     1,   80,   47;
     1,  193,  513,   1;

Ludovic Schwob, Illustration of T(6,k), 1<=k<=3
Wikipedia, Turning number

Row sums give A295264(n+1).

Cf. A029760, A030110, A342968.

B(n,m,x)={
  local(Cache=Map());
  my(recurse(k,p,q,b) = my(hk=[k,p,q,b], z); if(!mapisdefined(Cache, hk, &z),
  z = if(k==0, q>p && q>m, sum(j=1, n-(q-p)%n, my(r=(q+j)%n); if(!bittest(b,r), if(rAndrew Howroyd, May 15 2021

T(n,1)=1 and T(2*n-1,n)=1 for all n>=1: the only solutions are the polygons with respective Schläfli symbols {n+2} and {2*n+1/n}.
T(n,2) = A030110(n+1) for all n>=1.
T(2*n,n-1) = A029760(n-2) for all n>=2.

a(31)-a(49) from Andrew Howroyd, May 15 2021

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.

Original entry on oeis.org

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

Ludovic Schwob, Aug 21 2023

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).

			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;

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

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

Cf. A295264, A342968, A008292.

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

A371611 Number of 2*n-sided cycles with the property that one makes the same number of left and right turns while following its edges.

Original entry on oeis.org

4, 36, 1240, 73240, 7171176, 1016813448, 198480110880, 50752206180576, 16460660622560680, 6595414427636900536, 3198428240666246044704, 1845848150787599809368856, 1250049326783769438348496480, 981653074459964543314138858320

Offset: 2

Ludovic Schwob, Mar 29 2024

nonn

Ludovic Schwob, Illustration of a(3)

Cf. A365094, A295264,

a(n) is always divisible by 2*n, because the considered cycles cannot have rotational symmetry.

cp's OEIS Frontend

A343257 Triangle read by rows: T(n,k) is the number of n+2-sided polygons whose points lie on a circle and with the property that one makes k turns on itself, always in the same direction (right or left) while following its edges, 1 <= k <= ceiling(n/2).

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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.

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Examples

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Formula

A371611 Number of 2*n-sided cycles with the property that one makes the same number of left and right turns while following its edges.

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Author

Keywords

Comments

Links

Crossrefs

Formula