A396266 - OEIS

A396266

Area of the smallest simple polygon (by area) which can be triangulated into n different primitive Pythagorean triangles.

6, 36, 246, 792, 2178, 7824, 15846, 26220, 36930, 49584, 75390, 119964, 145770, 185964, 260010, 333792, 344946, 370752, 444798, 558912, 675906, 829416, 1001718, 1175520, 1289634, 1406628, 1676502, 1967268, 2180238, 2254284, 2371278, 2524788, 2697090, 2910960

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OFFSET

1,1

COMMENTS

Here polygons are strictly simple, such that all vertices lie on the perimeter of the polygon, no vertex touches more than 2 line segments on the perimeter (as in a degenerate simple polygon, with a fold or enclosing a hole), and no vertex is in the middle of a straightaway.

Variations of this sequence that relax some or all of these conditions still produce many of the same terms, including all terms up to and including n = 26. For n = 27, the area of the smallest self-intersecting polygon that can be constructed from 27 different primitive Pythagorean triangles adjoined at equal length edges is 1560138, which is smaller than a(27) (1676502).

For most polygons, the smallest constituent triangle is 3,4,5; n = 6 is the only known exception, with 8,15,17 as its smallest triangle. It is unknown if this is the only such case.

LINKS

Table of n, a(n) for n=1..34.

Charles L. Hohn, Example illustration of a(34)

Charles L. Hohn, PARI function for generating A396266 and related sequences, 2026.

EXAMPLE

a(1) = 6: a 3,4,5 triangle.

a(2) = 36: a 3,4,5 and a 5,12,13 triangle adjoined at their sides of length 5, in any orientation.

a(3) = 246: 3,4,5, 5,12,13, and 12,35,37, in any orientation.

PROG

(PARI) \\ see Hohn link.

CROSSREFS

Cf. A024406 (areas of primitive Pythagorean triangles).