A298860 Primitive cyclic quadrilaterals with integer area.

1, 3, 6, 8, 18, 12, 1, 5, 5, 7, 18, 16, 1, 2, 8, 9, 20, 12, 1, 5, 5, 9, 20, 15, 1, 4, 7, 8, 20, 18, 2, 5, 5, 8, 20, 20, 2, 5, 5, 10, 22, 18, 3, 5, 5, 9, 22, 24, 2, 4, 7, 11, 24, 20, 3, 5, 5, 11, 24, 21, 4, 5, 5, 10, 24, 28, 2, 6, 7, 9, 24, 30, 4, 5, 5, 12, 26, 24, 3, 4, 8, 11, 26, 30, 4, 5, 7, 10, 26, 36, 2, 5, 10, 11, 28, 36, 1, 7, 8, 14, 30, 28, 1, 8, 9, 12, 30, 42

Offset

1,2

Comments

Entries are listed as sextuples: (a,b,c,d), Perimeter, Area. They are ordered first by perimeter, second by area, third by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.

We observe that the number of odd integers in any quadruple is always an even number.

Links

Table of n, a(n) for n=1..108.

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

Wikipedia, Cyclic quadrilateral

Example

The first row of the table gives sidelengths (a,b,c,d)=(1,3,6,8) with perimeter=18 and area=12. Thus:
  a b c  d Perim Area
  = = = == ===== ====
  1 3 6  8   18   12
  1 5 5  7   18   16
  1 2 8  9   20   12
  1 5 5  9   20   15
  1 4 7  8   20   18
  2 5 5  8   20   20
  2 5 5 10   22   18
  3 5 5  9   22   24
  2 4 7 11   24   20
  3 5 5 11   24   21
  4 5 5 10   24   28
  etc.

Crossrefs

Cf. A298907, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.

Sequence in context: A123208 A179652 A253855 * A068979 A101060 A278486

Adjacent sequences: A298857 A298858 A298859 * A298861 A298862 A298863

Keyword

nonn,tabf

Author

Gregory Gerard Wojnar, Jan 27 2018

Status

approved