OFFSET
1,6
COMMENTS
Starting with a triangle partitioned into 3 trapezoids, there are two mutually exclusive ways of obtaining a triangle partitioned into 4 trapezoids.
Category A: Add a trapezoid to the base of the triangle.
Category B: Split one of the trapezoids parallel to its base.
In category A, there will be one trapezoid whose longest base is n, but in category B every trapezoid will have its longest base less than n.
The two constructions are illustrated below:
. . . .
/ \ A / \ / \ B / \
/__/\ => /__/\ /__/\ => /__/\
/___\_\ /___\_\ / \ \ /___\ \
/_______\ /_____\_\ /_____\_\
The number of cases in Category A appears to match A014125 up to offset. - Sean A. Irvine, Dec 16 2025
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..200 (terms 1..75 from Janaka Rodrigo)
Janaka Rodrigo, Python Code for Category B of A389392
Janaka Rodrigo, Python Code for Category A of A389392
FORMULA
Conjectured g.f.: x^5 * (1+9*x+19*x^2+29*x^3+27*x^4+14*x^5+3*x^6-6*x^7) / ((x^2+1) * (x+1)^2 * (x^2+x+1)^2 * (x-1)^4). - Sean A. Irvine, Oct 15 2025
EXAMPLE
A trapezoid whose base angles are 60 degrees with longer base b and legs s is denoted by {b X s} here.
Sets belong to A(6),
{{2 X 1}, {3 X 1}, {3 X 2}, {6 X 2}},
{{2 X 1}, {4 X 1}, {6 X 1}, {4 X 3}},
{{3 X 1}, {3 X 2}, {6 X 1}, {4 X 2}}.
Sets belong to B(6),
{{2 X 1}, {3 X 1}, {4 X 2}, {5 X 2}},
{{2 X 1}, {3 X 1}, {4 X 1}, {5 X 3}},
{{2 X 1}, {3 X 2}, {5 X 1}, {5 X 2}},
{{3 X 1}, {4 X 1}, {3 X 2}, {5 X 2}},
{{3 X 1}, {4 X 1}, {5 X 1}, {4 X 3}},
{{4 X 1}, {3 X 2}, {5 X 1}, {4 X 2}}.
Therefore a(6) = |A(6)| + |B(6)| = 9.
CROSSREFS
KEYWORD
nonn
AUTHOR
Janaka Rodrigo, Oct 02 2025
STATUS
approved