A347750 Number of intersection points when every pair of vertices of a row of n adjacent congruent rectangles are joined by an infinite line.

0, 5, 17, 57, 133, 297, 525, 925, 1477, 2289, 3277, 4701, 6437, 8805, 11541, 14917, 18869, 23893, 29509, 36473, 44349, 53545, 63605, 75629, 88901, 104325, 120981, 139913, 160581, 184409, 209885, 238989, 270525, 305413, 342413, 383301, 426949, 475757, 527205, 583261, 642821, 708717, 777829

Offset: 0

Author

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021

Keywords

• nonn

Examples

			a(1) = 5 as connecting the four vertices of a single rectangle forms one new vertex inside the rectangle, giving a total of 4 + 1 = 5 total intersection points.
a(2) = 17 as connecting the six vertices of two adjacent rectangles forms seven vertices inside the rectangles while also forming four vertices outside the rectangles. The total number of intersection points is then 6 + 7 + 4 = 17.
See the linked images for further examples.
		

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.

Crossrefs

Cf. A344993 (number of polygons), A347751 (number of edges), A159065, A331755, A092275 (number of intersections resp. inside the rectangles, on or inside them, above them).

Formula

a(n) = A347751(n) - A344993(n) + 1.

It seems that a(n) = 2 * A159065(n+1) + 3 for n>0. - Andrei Zabolotskii, Jul 03 2025