A357058 Number of regions in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

1, 5, 17, 37, 65, 93, 145, 181, 257, 309, 401, 457, 577, 653, 785, 869, 1025, 1109, 1297, 1413, 1601, 1725, 1937, 2041, 2305, 2453, 2705, 2861, 3137, 3289, 3601, 3765, 4089, 4293, 4625, 4801, 5185, 5405, 5769, 5993, 6401, 6605, 7057, 7309, 7737, 8013, 8465, 8673, 9217, 9477, 9993, 10309

Offset: 0

Author

Scott R. Shannon, Sep 10 2022

Keywords

• nonn

Comments

The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .

Links

• Scott R. Shannon, Image for n = 1.
• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 5. This is the first term that forms squares with non-simple intersections.
• Scott R. Shannon, Image for n = 10.
• Scott R. Shannon, Image for n = 32. This is the first term with n mod 2 = 0 that forms squares with non-simple intersections.
• Scott R. Shannon, Image for n = 200.

Crossrefs

Cf. A357060 (vertices), A357061 (edges), A108914, A355838, A355798, A356984 (triangle).

Formula

a(n) = A357061(n) - A357060 (n) + 1 by Euler's formula.

Conjecture: a(n) = 4*n^2 + 1 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.