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

0,2

Comments

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

Links

Table of n, a(n) for n=0..51.

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.

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.

Crossrefs

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

Sequence in context: A273538 A273212 A273274 * A273250 A053755 A162373

Adjacent sequences: A357055 A357056 A357057 * A357059 A357060 A357061

Keyword

nonn

Author

Scott R. Shannon, Sep 10 2022

Status

approved