A391452 - OEIS

A391452

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 without partitioning a triangle into 3 element sets of trapezoids.

%I #12 Dec 16 2025 20:53:17

%S 0,0,0,0,0,1,6,25,55,115,197,324,491,727,1020,1407,1882,2476,3181,

%T 4049,5054,6257,7644,9263,11103,13230,15614,18334,21373,24792,28572,

%U 32807,37453,42608,48252,54458,61209,68613,76617,85342,94760,104962,115923,127774,140453,154095,168674,184293,200924,218717,237596,257724,279067,301741,325715

%N 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 without partitioning a triangle into 3 element sets of trapezoids.

%C This can be done under two mutually exclusive categories:

%C Category-1: Let P and Q be two points inside the triangle ABC such that PQ is in the same direction of AB. Points D, E are marked on AB such that PD is parallel to CA and QE is parallel to CB. Point F is marked on BC such that QF is parallel to AC and point S is marked on AC such that PS is parallel to BC. Finally point R is marked on QF such that SR is parallel to AB. The five trapezoids are PDEQ, ADPS, QEBF, PQRS and SRFC.

%C Category-2: Let P be a point inside the triangle ABC. Points D,E are marked on AB,BC respectively such that PD is parallel to CB and PE is parallel to AC. Points N,L are marked on AC,PD respectively such that NL is parallel to AB and points M,G are marked on NL,CE respectively such that MG is parallel to AC. Finally point F is marked on MG such that FP is parallel to AB. The five trapezoids are PDBE, ADLN, MNCG, PEGF, MLPF.

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

%H Sean A. Irvine, <a href="/A391452/b391452.txt">Table of n, a(n) for n = 1..150</a>

%H Janaka Rodrigo, <a href="/A391452/a391452.txt">Python Code for the Union of two categories</a>

%e n = 7 has 6 sets of trapezoids:

%e Category-1

%e {{2 X 1}, {3 X 1}, {3 X 2}, {4 X 2}, {5 X 3}},

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

%e Category-2

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

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

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

%e {{3 X 1}, {3 X 2}, {5 X 1}, {6 X 1}, {5 X 2}}.

%e Therefore a(7) = 6.

%Y Cf. A391039, A391203, A391204, A391453.

%K nonn

%O 1,7

%A _Janaka Rodrigo_, Dec 09 2025