A391453 - OEIS

A391453

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, where one trapezoid has a larger base of n units.

0, 0, 0, 0, 0, 1, 10, 30, 70, 136, 237, 386, 593, 873, 1238, 1706, 2295, 3024, 3906, 4974, 6243, 7737, 9483, 11512, 13840, 16512, 19546, 22978, 26844, 31181, 36008, 41388, 47342, 53912, 61143, 69084, 77757, 87237, 97548, 108744, 120879, 134008

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OFFSET

1,7

COMMENTS

This can be done under two mutually exclusive categories:

Category-1: Let P be a point inside the triangle ABC. Points D, F, M are marked on the side AC in the given order and points E,G,L are marked on the side BC in the given order, such that DE, FG and PL are parallel to AB, then PM is parallel to BC and NP is parallel to AC. The five trapezoids are ABED, DEGF, PNGL, PLCM and PMFN.

Category-2: Let P be a point inside the triangle ABC. Points D,M are marked on the side AC in the given order, points E, H, L are marked on the side BC in the given order, such that DE,PL are parallel to AB and PM is parallel to BC. Point N is marked on DE such that PN is parallel to CA and point G is marked on PN such that GH is parallel to AB. The five trapezoids are ABED, LCMP, PMDN, PLHG, GHEN.

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

LINKS

Sean A. Irvine, Table of n, a(n) for n = 1..150

Janaka Rodrigo, Python Code for Union of two categories

EXAMPLE

n = 7 has 10 sets of trapezoids:

Category-1

{{2 X 1}, {3 X 1}, {3 X 2}, {5 X 1}, {7 X 2}},

{{2 X 1}, {3 X 1}, {3 X 2}, {7 X 1}, {6 X 2}},

{{2 X 1}, {4 X 1}, {6 X 1}, {7 X 1}, {4 X 3}},

{{3 X 1}, {3 X 2}, {6 X 1}, {4 X 2}, {7 X 1}}.

Category-2