A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

1, 5, 18, 52, 125, 257, 486, 832, 1333, 2027, 3048, 4304, 6057, 8167, 10749, 13929, 18058, 22664, 28533, 34981, 42519, 51425, 62118, 73473, 86768, 101902, 118695, 137138, 159147, 181752, 208813, 237209, 268614, 303718, 340882, 380811, 427540, 477134, 530047

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

Keywords

• nonn

Comments

The terms are from numeric computation - no formula for a(n) is currently known.

Equivalently, this is also the number of regions formed when all the integer points along the x and y axes with 0 <= x <= n and 0 <= y <= n are joined by straight line segments.

If instead one takes points on the x and y axes with coordinates 1, 1/2, 1/3, 1/4, ..., 1/n, 0, and joins them all by line segments, the resulting figure contains only triangles and quadrilaterals, and the number of regions is given by A332358 (and more generally by A332357 if there are m+1 such points on the x axis and n+1 such points on the y axis).

Links

• Lars Blomberg, Table of n, a(n) for n = 1..70
• Scott R. Shannon, Illustration for n = 2.
• Scott R. Shannon, Illustration for n = 3.
• Scott R. Shannon, Illustration for n = 4.
• Scott R. Shannon, Illustration for n = 5.
• Scott R. Shannon, Illustration for n = 6.
• Scott R. Shannon, Illustration for n = 8.
• Scott R. Shannon, Illustration for n = 10.
• Scott R. Shannon, Illustration for n = 12.
• Scott R. Shannon, Illustration for n = 15.
• Scott R. Shannon, Illustration for n = 5 with random distance-based coloring.
• Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
• Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.

Crossrefs

Cf. A333025 (n-gons), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

Extensions

a(16) and beyond from Lars Blomberg, May 26 2020