A340859 a(n) is the number of isosceles integer trapezoids (up to congruence) with integer side lengths a,c,b=d with n=Max(a,b,c) and integer diagonals e=f.

0, 0, 0, 1, 1, 1, 2, 5, 6, 3, 3, 9, 6, 5, 10, 20, 9, 10, 8, 21, 18, 10, 10, 37, 21, 12, 24, 31, 14, 26, 17, 55, 32, 20, 36, 54, 22, 20, 39, 74, 24, 40, 26, 58, 59, 24, 26, 113, 47, 41, 54, 69, 33, 51, 61, 111, 65, 35, 39, 124, 38, 39, 88, 145, 79

Offset

1,7

Comments

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.

Without loss of generality we assume b=d and for the parallel sides c < a. e and f are uniquely determined by e = f = sqrt((c(a^2-b^2) + a(b^2-c^2))/(a-c)). The smallest possible isosceles trapezoid has side lengths a=4, c=3, b=d=2 and diagonals e=f=4.

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&&#[[2]]==#[[4]]&]//Length, {n, 1, 65}]

Crossrefs

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340860 for non-isosceles trapezoids.

Sequence in context: A165501 A274614 A353771 * A336817 A340858 A309364

Adjacent sequences: A340856 A340857 A340858 * A340860 A340861 A340862

Keyword

nonn

Author

Herbert Kociemba, Jan 24 2021

Status

approved