OFFSET
1,13
COMMENTS
By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
Without loss of generality we assume for the parallel sides c < a and for the diagonals f < e. e and f are uniquely determined by e = sqrt((c(a^2-b^2) + a(d^2-c^2))/(a-c)) and f = sqrt((c(a^2-d^2) + a(b^2-c^2))/(a-c)).
The smallest possible trapezoid which is not isosceles has side lengths a=8, b=9, c=3, d=11 and diagonals e=13 and f=9.
EXAMPLE
a(34)=2 because up to congruence there are exactly two trapezoids which are not isosceles:
a=32, b=26, c=22, d=34 and e=54, f=18;
a=34, b=11, c=32, d=12 and e=40, f=29.
MATHEMATICA
n=65; list={};
For[a=1, a<=n, a++,
For[c=1, c<a, c++,
For[d=Floor[(a-c)/2]+1, d<=n, d++,
For[b=1, b<=n, b++,
se=c(a^2-b^2)+a(d^2-c^2); sf=c(a^2-d^2)+a(b^2-c^2);
If[se<=0||sf>se, Break[]]; If[sf<=0, Continue[]];
e=Sqrt[se/(a-c)]; f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b, AppendTo[list, {a, b, c, d, e, f}]]]]]]
Table[Select[list, Max[#[[1]], #[[2]], #[[3]], #[[4]]]==n&&#[[2]]!=#[[4]]&]//Length, {n, 1, 65}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Jan 24 2021
STATUS
approved