A391498 a(n) is the number of 5 element sets of distinct integer-sided trapezoids whose base angles are 60 degrees that fill an equilateral triangular grid of side n units.

0, 0, 0, 0, 0, 2, 20, 83, 199, 422, 746, 1255, 1940, 2900, 4137, 5753, 7770, 10276, 13334, 17026, 21416, 26592, 32680, 39689, 47831, 57085, 67669, 79613, 93138, 108129, 125105, 143792, 164592, 187468, 212845, 240368, 270877, 303817, 339919, 378972, 421563, 467085, 516915, 570024, 627439, 688829, 755085

Offset

1,6

Comments

This is the union of eight categories AA1, AA2, BB1, BB2, CC1, CC2, CC3 and DD1. Category DD1 describes how the triangle is filled by five distinct trapezoids by splitting each of the two trapezoids parallel to the larger base into two trapezoids of the three element sets.

A trapezoid whose base angles are 60 degrees with larger base b and legs s is denoted by {b X s} here.

Links

Table of n, a(n) for n=1..47.

Janaka Rodrigo, Python code for the category DD1

Janaka Rodrigo, Formation of the eight categories

Janaka Rodrigo, Example sets for n = 7

Janaka Rodrigo, Illustration of a(7)

Janaka Rodrigo, Python Code for the union of eight categories of five trapezoid sets

Example

For n = 7, there are 27 solutions over all categories, with only 20 distinct sets of distinct trapezoids. Therefore, a(7) = 20. See links for illustrations of the 27 solutions.

Crossrefs

Cf. A391204 (union of CC1, CC2, CC3), A389392 (4 element sets of trapezoids).

Sequence in context: A135188 A161007 A394528 * A331760 A098077 A279264

Adjacent sequences: A391495 A391496 A391497 * A391499 A391500 A391501

Keyword

nonn

Author

Janaka Rodrigo, Dec 11 2025

Status

approved