A340860 a(n) is the number of non-isosceles integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 2, 1, 0, 1, 1, 0, 4, 4, 9, 5, 9, 11, 7, 4, 5, 3, 11, 13, 2, 7, 3, 3, 5, 9, 8, 3, 6, 9, 12, 10, 19, 8, 23, 16, 16, 18, 21, 13, 25, 19, 32, 26, 7, 7, 25, 16, 8, 27, 59, 26

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.

Links

Table of n, a(n) for n=1..65.

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

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340859 for isosceles trapezoids.

Sequence in context: A193233 A145878 A195662 * A112606 A108512 A054503

Adjacent sequences: A340857 A340858 A340859 * A340861 A340862 A340863

Keyword

nonn

Author

Herbert Kociemba, Jan 24 2021

Status

approved