

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.


6



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
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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^2b^2) + a(d^2c^2))/(ac)) and f = sqrt((c(a^2d^2) + a(b^2c^2))/(ac)).
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[(ac)/2]+1, d<=n, d++,
For[b=1, b<=n, b++,
se=c(a^2b^2)+a(d^2c^2); sf=c(a^2d^2)+a(b^2c^2);
If[se<=0sf>se, Break[]]; If[sf<=0, Continue[]];
e=Sqrt[se/(ac)]; f=Sqrt[sf/(ac)];
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 nonisosceles trapezoids.
Sequence in context: A274614 A340859 A336817 * A309364 A062825 A154925
Adjacent sequences: A340855 A340856 A340857 * A340859 A340860 A340861


KEYWORD

nonn


AUTHOR

Herbert Kociemba, Jan 24 2021


STATUS

approved



