login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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^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 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[(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&]//Length, {n, 1, 65}]

CROSSREFS

Cf. A224931 for parallelograms, A340859 and A340860 for isosceles and non-isosceles 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)