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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. 2
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, 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, 16, 8, 27, 59, 26 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

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 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(34)=2 because up to congruence there are exactly two trapezoids which are not isosceles:

a=32, b=26, c=22, d=34 and e=54, f=18;

a=34, b=11, c=32, d=12 and e=40, f=29.

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 A340859 for isosceles trapezoids.

Sequence in context: A193233 A145878 A195662 * A112606 A108512 A054503

Adjacent sequences:  A340857 A340858 A340859 * A340861 A340862 A340863

KEYWORD

nonn

AUTHOR

Herbert Kociemba, Jan 24 2021

STATUS

approved

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Last modified May 17 12:43 EDT 2021. Contains 343971 sequences. (Running on oeis4.)