A391204 - OEIS

A391204

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 with a trapezoid filled by 3 trapezoids.

0, 0, 0, 0, 0, 0, 6, 24, 53, 115, 200, 332, 520, 782, 1102, 1553, 2093, 2769, 3605, 4623, 5795, 7248, 8899, 10818, 13066, 15639, 18500, 21858, 25564, 29714, 34416, 39636, 45319, 51775, 58769, 66424, 74918, 84161, 94081, 105115, 116922, 129645, 143550, 158481, 174354, 191729, 210150, 229791, 251023, 273594

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OFFSET

1,7

COMMENTS

Let P be a floating point inside of the triangle ABC such that PDBE,PECF, and PFAD are trapezoids whose base angles are 60 degrees. Where the points D,E,F lie on the sides AB,BC and CA respectively. Partitioning the trapezoid PECF into three distinct trapezoids is done under three mutually exclusive ways:

Category-C1: Select a point M on PF and a point N on CE such that MN is parallel to PE. Then select a point G on MP and a point H on NE such that GH is parallel to FC. Three trapezoids are PEHG, GHNM and MNCF.

Category-C2: Select a point M on PF and a point N on CE such that MN is parallel to PE. Then select a point G on PE and a point H on MN such that GH is parallel to AC. Three trapezoids are PGHM, GENH and MNCF.

Category-C3: Select a point G on PE and a point H on FC such that GH is parallel to CE. Then select a point M on GE and a point N on HC such that MN is parallel to CE. Three trapezoids are PGHF, GMNH and MECN.

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..50.

Janaka Rodrigo, Python Code for Union of C1,C2,C3

EXAMPLE

n = 7 has 6 sets of trapezoids:

C1 has 0 sets.

C2 has 1 set,

{{2 X 1}, {3 X 1}, {3 X 2}, {5 X 1}, {5 X 4}}.

C3 has 5 sets,

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

{{2 X 1}, {3 X 1}, {5 X 1}, {4 X 2}, {6 X 2}},