A334763 Ceiling of circumradius of triangle whose sides are consecutive Ulam numbers (A002858).

3, 4, 5, 6, 7, 9, 10, 14, 15, 19, 21, 24, 26, 29, 31, 34, 37, 40, 43, 45, 48, 52, 55, 58, 60, 63, 68, 72, 77, 80, 84, 87, 93, 99, 103, 104, 107, 110, 115, 118, 123, 126, 131, 134, 138, 139, 142, 146, 149, 153, 158, 168, 176, 182, 185, 190, 194, 200, 204, 208

Offset

2,1

Comments

It has been proved that three consecutive Ulam numbers U(n) for n > 1 satisfy the triangle inequality. See Wikipedia link below. Consequently it is possible to create n-gons using n consecutive Ulam numbers. The sequence starts at offset 2 because using the first Ulam number generates a triangle with sides (1,2,3) that is degenerate with infinite circumradius.

Conjecture: Triangles whose sides are consecutive Ulam numbers are acute apart from (1,2,3), (2,3,4), (3,4,6), (4,6,8), (6,8,11) and (16,18,26).

Links

Table of n, a(n) for n=2..61.

Eric Weisstein's World of Mathematics, Circumradius.

Eric Weisstein's World of Mathematics, Ulam Sequence.

Wikipedia, Ulam number.

Formula

Circumradius of a triangle with sides a, b, c is given by R = a*b*c/(4A) where the Area A is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) and where s = (a+b+c)/2.

Example

a(2)=3 because a triangle with sides 2,3,4 has area = (1/4)*sqrt((2+3+4)(2+3-4)(2-3+4)(-2+3+4)) = 2.904... and circumradius = 2*3*4/(4A) = 2.065...

Mathematica

lst1=ReadList["https://oeis.org/A002858/b002858.txt", {Number, Number}]; lst={}; Do[{a, b, c}={lst1[[n]][[2]], lst1[[n+1]][[2]], lst1[[n+2]][[2]]}; s=(a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; R=a*b*c/(4 A); AppendTo[lst, Ceiling@R], {n, 2, 100}]; lst

Crossrefs

Cf. A002858, A331676.

Sequence in context: A298006 A026438 A026442 * A307712 A048869 A039051

Adjacent sequences: A334760 A334761 A334762 * A334764 A334765 A334766

Keyword

nonn

Author

Frank M Jackson, May 10 2020

Status

approved