A358574 Triangle read by rows: T(n,k) is the number of vertices 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.

8, 12, 20, 12, 16, 16, 16, 64, 16, 20, 36, 20, 68, 36, 100, 20, 24, 36, 24, 24, 44, 144, 29, 144, 24, 28, 28, 28, 92, 28, 140, 28, 44, 76, 208, 28, 32, 44, 32, 84, 52, 32, 39, 240, 88, 292, 46, 296, 32, 36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36, 40, 40, 40, 80, 40, 164, 47, 40, 112, 364, 40, 436, 144, 88, 67, 472, 40

Offset

1,1

Comments

See A358556 for further details.

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) = 20.

Scott R. Shannon, Image for T(4,6) = 36.

Scott R. Shannon, Image for T(7,9) = 240.

Scott R. Shannon, Image for T(10,19) = 584.

Scott R. Shannon, Image for T(11,20) = 90.

Scott R. Shannon, Image for T(20,11) = 308.

Scott R. Shannon, Image for T(20,30) = 100.

Scott R. Shannon, Image for T(20,31) = 2220.

Formula

T(n,k) = A358627(n,k) - A358556(n,k) + 1 by Euler's formula.

T(n,2*n) = 4*(n + 1). The line cuts the square into two parts so no new vertices are created.

T(n,k) = 4*(n + 1) where k <= n and k|(4*n). Four lines cut across the square's corners so no new vertices are created.

Example

The table begins:
8;
12, 20, 12;
16, 16, 16, 64, 16;
20, 36, 20, 68, 36, 100, 20;
24, 36, 24, 24, 44, 144, 29, 144, 24;
28, 28, 28, 92, 28, 140, 28,  44,  76, 208, 28;
32, 44, 32, 84, 52,  32, 39, 240,  88, 292, 46, 296,  32;
36, 48, 36, 80, 52, 164, 36, 252, 100, 316, 52, 368, 124, 364, 36;
40, 40, 40, 80, 40, 164, 47,  40, 112, 364, 40, 436, 144,  88, 67, 472, 40;
.
.
See the attached file for more examples.

Crossrefs

Cf. A358556 (regions), A358627 (edges), A331452, A355798, A355838, A357058, A358407, A345459.

Sequence in context: A168670 A291758 A232867 * A084488 A337877 A211410

Adjacent sequences: A358571 A358572 A358573 * A358575 A358576 A358577

Keyword

nonn,tabf

Author

Scott R. Shannon, Nov 23 2022

Status

approved