A348143 Areas of integer-sided cyclic quadrilaterals whose area equals their perimeter.

16, 18, 20, 30

Offset

1,1

Comments

There are no further terms. Note that without the condition "integer-sided" there are other solutions, such as (1, 17/2, 17/2, 16) which has perimeter and area 34.

Links

Table of n, a(n) for n=1..4.

Ralph H. Buchholz and James A. MacDougall, Cyclic Polygons with Rational Sides and Area, 2001. JNT 128 (2008) 17-48

Example

The areas or perimeters 16, 18, 20, 30 pertain respectively to cyclic quadrilaterals with sides (4, 4, 4, 4), (3, 3, 6, 6), (2, 5, 5, 8), (5, 5, 6, 14).

Mathematica

lst={}; Do[s=(a+b+c+d)/2; If[s>a, (K=Sqrt[(s-a)(s-b)(s-c)(s-d)]; If[IntegerQ[K]&&K==2s, AppendTo[lst, Sort@{a, b, c, d}]])], {a, 1, 15}, {b, 1, a}, {c, 1, b}, {d, 1, c}]; lst

Crossrefs

Cf. A098030, A290451. First four terms of A161874.

Sequence in context: A273543 A274127 A328736 * A161874 A095953 A084800

Adjacent sequences: A348140 A348141 A348142 * A348144 A348145 A348146

Keyword

fini,full,nonn

Author

Frank M Jackson, Oct 02 2021

Status

approved