A355798 Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

1, 4, 24, 104, 316, 712, 1588, 2816, 4940, 7672, 12444, 16840, 25968, 34088, 46260, 61048, 82792, 98984, 133032, 156072, 196236, 239048, 298292, 334032, 417072, 483856, 570200, 649816, 786412, 850000, 1037628, 1145424, 1311536, 1485880, 1677660, 1828360, 2158192, 2357376, 2623604, 2852688

Offset: 1

Author

Scott R. Shannon, Jul 17 2022

Keywords

• nonn

Links

• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 7.
• Scott R. Shannon, Image for n = 8.
• Scott R. Shannon, Image for n = 16.
• Scott R. Shannon, Alternative version for n = 15. The lines joining the vertices that are exactly opposite are omitted, forming a more carpet-like image. The number of regions formed, 42224, is less than the sequence version, although more 5,6,7 and 8-gon regions are present.
• N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 20-21.

Crossrefs

Cf. A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

Formula

a(n) = A355800(n) - A355799(n) + 1 by Euler's formula.