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%I #12 Feb 22 2021 02:25:44
%S 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,
%T 18,31,20,66,45,22,43,57,25,25,48,82,27,46,35,70,69,43,34,136,63,57,
%U 72,90,46,76,80,143,91,42,46,149,54,47,115,204,105
%N 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.
%C By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
%C 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)).
%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.
%e 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.
%t n=65;list={};
%t For[a=1,a<=n,a++,
%t For[c=1,c<a,c++,
%t For[d=Floor[(a-c)/2]+1,d<=n,d++,
%t For[b=1,b<=n,b++,
%t se=c(a^2-b^2)+a(d^2-c^2);sf=c(a^2-d^2)+a(b^2-c^2);
%t If[se<=0||sf>se,Break[]];If[sf<=0,Continue[]];
%t e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
%t 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}]]]]]]
%t Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&]//Length,{n,1,65}]
%Y Cf. A224931 for parallelograms, A340859 and A340860 for isosceles and non-isosceles trapezoids.
%K nonn
%O 1,7
%A _Herbert Kociemba_, Jan 24 2021
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