A298907 Primitive cyclic quadrilaterals with integer area.

1, 3, 6, 8, 1, 5, 5, 7, 1, 2, 8, 9, 1, 5, 5, 9, 1, 4, 7, 8, 2, 5, 5, 8, 2, 5, 5, 10, 3, 5, 5, 9, 2, 4, 7, 11, 3, 5, 5, 11, 4, 5, 5, 10, 2, 6, 7, 9, 4, 5, 5, 12, 3, 4, 8, 11, 4, 5, 7, 10, 2, 5, 10, 11, 1, 7, 8, 14, 1, 8, 9, 12, 3, 7, 9, 11, 1, 6, 10, 15, 2, 7, 9, 14, 1, 7, 11, 13, 6, 7, 8, 11, 1, 10, 10, 13, 2, 9, 11, 12, 3, 6, 13, 14, 3, 8, 10, 15

Offset: 1

Author

Gregory Gerard Wojnar, Jan 28 2018

Keywords

• nonn
• tabf

Comments

Entries are listed as quadruples: (a,b,c,d). They are ordered first by perimeter, second by area, then 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.

It appears that the number of odd sidelengths in any quadruple is always 0, 2, or 4.

Examples

			We list here the early quadruplets, in parentheses, augmented by the associated perimeter and area to justify the ordering of the quadruplets:
(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
(2,  6,  7,  9)    24    30
etc.
		

Crossrefs

Cf. A298860, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.