A358556 Triangle read by rows: T(n,k) is the number of regions formed when n points are placed along each edge of a square that divide the edges into n+1 equal parts and a line is continuously drawn from the current point to that k points, 2 <= k <= 2*n, counterclockwise around the square until the starting point is again reached.

2, 5, 21, 2, 5, 5, 4, 61, 2, 5, 29, 5, 73, 25, 105, 2, 5, 25, 5, 5, 31, 141, 11, 157, 2, 5, 5, 5, 85, 5, 153, 4, 25, 61, 229, 2, 5, 25, 5, 73, 33, 5, 15, 245, 71, 297, 22, 317, 2, 5, 25, 5, 65, 29, 165, 5, 269, 81, 333, 25, 385, 109, 401, 2, 5, 5, 5, 61, 5, 153, 16, 5, 91, 377, 4, 449, 125, 61, 37, 509, 2

Offset

1,1

Comments

The starting point can be any of the 4*n points around the square as changing the starting point simply rotates and/or reflects the resulting pattern formed by the path to one of the four orthogonal directions around the square; this does not change the number of regions formed by the path.

The number of times the path formed by the line touches and leaves the edges of the square is lcm(4*n,k)/k. For k >= n this is the number of points in the star-shaped pattern formed by the path.

The table starts with k = 2 as T(n,1) = 5 for all values of n. The maximum k is 2*n as T(n,2*n + m) = T(n,2*n - m).

Links

Table of n, a(n) for n=1..81.

Scott R. Shannon, Table for n=1..50.

Scott R. Shannon, Image for T(2,3) = 21.

Scott R. Shannon, Image for T(4,6) = 25.

Scott R. Shannon, Image for T(7,9) = 245.

Scott R. Shannon, Image for T(10,19) = 629.

Scott R. Shannon, Image for T(11,20) = 55.

Scott R. Shannon, Image for T(20,11) = 269.

Scott R. Shannon, Image for T(20,30) = 25.

Scott R. Shannon, Image for T(20,31) = 2277.

Scott R. Shannon, Image for T(50,61) = 11933.

Formula

T(n,k) = A358627(n,k) - A358574(n,k) + 1 by Euler's formula.

T(n,2*n) = 2. The line cuts the square into two parts.

T(n,k) = 5 where n >= 2, k <= n, and k|(4*n). Four lines cut across the square's corners so four additional triangles are created.

Example

The table begins:
2;
5, 21, 2;
5,  5  4, 61,  2;
5, 29, 5, 73, 25, 105,  2;
5, 25, 5,  5, 31, 141, 11, 157,  2;
5,  5, 5, 85,  5, 153,  4,  25, 61, 229,  2;
5, 25, 5, 73, 33,   5, 15, 245, 71, 297, 22, 317,   2;
5, 25, 5, 65, 29, 165,  5, 269, 81, 333, 25, 385, 109, 401,  2;
5,  5, 5, 61,  5, 153, 16,   5, 91, 377,  4, 449, 125,  61, 37, 509,   2;
5, 25, 5,  5, 25, 137,  5, 285,  5, 385, 31, 501, 141,  25, 11, 613, 169, 629, 2;
.
.
See the attached file for more examples.

Crossrefs

Cf. A358574 (vertices), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

Sequence in context: A003163 A088498 A240147 * A293687 A190117 A230656

Adjacent sequences: A358553 A358554 A358555 * A358557 A358558 A358559

Keyword

nonn,tabf

Author

Scott R. Shannon, Nov 22 2022

Status

approved