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

1,2

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.

Sequence in context: A272512 A257055 A036893 * A125641 A006479 A127983

Adjacent sequences: A332950 A332951 A332952 * A332954 A332955 A332956

Keyword

nonn

Author

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

Extensions

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

Status

approved