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 A334763 Ceiling of circumradius of triangle whose sides are consecutive Ulam numbers (A002858). 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 12 10:52 EDT 2021. Contains 342920 sequences. (Running on oeis4.)