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A276072 Radius of the outer Soddy circle (or the longest side) of primitive Heronian triangles whose outer Soddy center lies on its longest side. 1
6, 21, 52, 95, 105, 186, 259, 273, 301, 392, 456, 549, 603, 657, 790, 910, 1001, 1023, 1067, 1133, 1221, 1308, 1596, 1651, 1677, 1729, 1807, 1911, 2041, 2114, 2282, 2535, 2562, 2715, 2985, 3088, 3165, 3216, 3472, 3689, 3723, 3791, 3856, 3893, 4029, 4199, 4403, 4446, 4641, 4662 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It has been shown that if the outer Soddy circle center of a triangle lies on one of its sides then this side is the longest and is equal in length to both the outer Soddy radius and the excircle radius associated with the vertex opposite the longest side. Furthermore, the sum of the tangents of the triangle's half angles opposite the two smaller sides equals 1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

F. M. Jackson, S. Takhaev, Heronian Triangles of Class J: Congruent Incircles Cevian Perspective, IJCDM, 1/3 (2016), 1-8.

EXAMPLE

The triangle ABC with sides {a, b, c} = {21, 20, 13} has an outer Soddy radius and an exradius (opp. vertex A) equal to 21. Its area is 126. The outer Soddy center lies on BC (= 21) at the tangential point with the A excircle at length 15 from B.

MATHEMATICA

a=2n(m^2+3n^2); b=(m+n)(m^2-2m*n+5n^2); c=-(m-n)(m^2+2m*n+5^2); lst={}; Do[If[GCD[m, n]==1, AppendTo[lst, a/GCD[a, b, c]]], {n, 1, 100}, {m, 0, n-1}]; Take[Sort@lst, 50] (* Corrected by Frank M Jackson, May 12 2017 *)

CROSSREFS

Sequence in context: A097124 A309568 A244906 * A135454 A069778 A015644

Adjacent sequences:  A276069 A276070 A276071 * A276073 A276074 A276075

KEYWORD

nonn

AUTHOR

Frank M Jackson, Stalislav Takhaev, Sep 10 2016

EXTENSIONS

Trailing terms corrected by Frank M Jackson, May 12 2017

B-file corrected by Giovanni Resta, May 13 2017

STATUS

approved

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Last modified July 26 01:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)