OFFSET
1,1
COMMENTS
The first cyclic quadrilateral in this sequence with sides (2,3,5,7) is analogous to an obtuse triangle in that the circumcenter does not lie within the bounds of the quadrilateral. Thereafter, the quadrilaterals have circumcenters that lie within the bounds of the quadrilateral.
LINKS
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral.
Wikipedia, Cyclic quadrilateral.
Wikipedia, Prime triplet.
FORMULA
The circumradius R of a cyclic quadrilateral with sides a, b, c, d is given by the Parameshvara's circumradius formula R = sqrt((ab+cd)(ac+bd)(ad+bc)/((s-a)(s-b)(s-c)(s-d)) where s = (a+b+c+d)/2.
EXAMPLE
a(2)=6 because a cyclic quadrilateral with sides (3,5,7,11) has circumradius = 5.56365...
MATHEMATICA
lst = {}; Do[{a, b, c, d}={Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}; s=(a+b+c+d)/2; R=Sqrt[(a*b+c*d)(a*c+b*d)(a*d+b*c)/((s-a)(s-b)(s-c)(s-d))]/4; AppendTo[lst, Ceiling@R], {n, 1, 100}]; lst
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jun 21 2021
STATUS
approved