A340858 a(n) is the number of 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, 1, 1, 1, 2, 5, 6, 3, 4, 9, 9, 7, 10, 22, 10, 10, 9, 22, 18, 14, 14, 46, 26, 21, 35, 38, 18, 31, 20, 66, 45, 22, 43, 57, 25, 25, 48, 82, 27, 46, 35, 70, 69, 43, 34, 136, 63, 57, 72, 90, 46, 76, 80, 143, 91, 42, 46, 149, 54, 47, 115, 204, 105

Offset

1,7

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 has side lengths a=4, c=3, b=d=2 and diagonals e=f=4. 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(7)=2 because there are two possible trapezoids: a=5, c=3, b=d=7, e=f=8 and a=7, c=4, b=d=6, e=f=8.

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&]//Length, {n, 1, 65}]

Crossrefs

Cf. A224931 for parallelograms, A340859 and A340860 for isosceles and non-isosceles trapezoids.

Sequence in context: A353771 A340859 A336817 * A309364 A062825 A154925

Adjacent sequences: A340855 A340856 A340857 * A340859 A340860 A340861

Keyword

nonn

Author

Herbert Kociemba, Jan 24 2021

Status

approved