cp's OEIS Frontend

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.

%I A340860 #11 Feb 22 2021 02:25:34
%S A340860 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,
%T A340860 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,
%U A340860 16,8,27,59,26
%N 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.
%C A340860 By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
%C A340860 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 A340860 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 A340860 a(34)=2 because up to congruence there are exactly two trapezoids which are not isosceles:
%e A340860 a=32, b=26, c=22, d=34 and e=54, f=18;
%e A340860 a=34, b=11, c=32, d=12 and e=40, f=29.
%t A340860 n=65;list={};
%t A340860 For[a=1,a<=n,a++,
%t A340860 For[c=1,c<a,c++,
%t A340860 For[d=Floor[(a-c)/2]+1,d<=n,d++,
%t A340860 For[b=1,b<=n,b++,
%t A340860 se=c(a^2-b^2)+a(d^2-c^2);sf=c(a^2-d^2)+a(b^2-c^2);
%t A340860 If[se<=0||sf>se,Break[]];If[sf<=0,Continue[]];
%t A340860 e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
%t A340860 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 A340860 Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&&#[[2]]!=#[[4]]&]//Length,{n,1,65}]
%Y A340860 Cf. A224931 for parallelograms, A340858 for general trapezoids and A340859 for isosceles trapezoids.
%K A340860 nonn
%O A340860 1,13
%A A340860 _Herbert Kociemba_, Jan 24 2021