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%I #23 Mar 02 2015 16:08:46
%S 2,6,3,12,8,4,20,15,10,5,30,24,18,12,6,42,35,28,21,14,7,56,48,40,32,
%T 24,16,8,72,63,54,45,36,27,18,9,90,80,70,60,50,40,30,20,10,110,99,88,
%U 77,66,55,44,33,22,11,132,120,108,96,84,72,60,48,36,24,12,156,143,130,117
%N Radius of A-excircle of Pythagorean triangle with a = (n+1)^2 - m^2, b = 2*(n+1)*m and c = (n+1)^2 + m^2.
%C From _Wolfdieter Lang_, Dec 03 2014: (Start)
%C For excircles and their radii see the _Eric W. Weisstein_ links. Here the circle radius with center J_A is considered.
%C Note that not all Pythagorean triangles are covered, e.g., the nonprimitive one (9, 12, 15) does not appear. However, the nonprimitive one (8, 6, 10) does appear as (n, m) = (2, 1). (End)
%C This triangle T appears also in the problem of finding all positive integer solutions for a and b of the general Fibonacci sequence F(a,b;k+1) = a*F(a,b;k) + b*F(a,b;k-1) (with some inputs F(a,b;0) and F(a,b;1)) such that the limit r = r(a,b) = F(a,b;k+1)/F(a,b;k) for k -> infinity becomes a positive integer r = (a + sqrt(a^2 + 4*b))/2. Namely, for any a = m >= 1 there are infinitely many b solutions b = T(n,m) = (n+1)*(n+1-m) for n >= m. The limit is r(a,b) = n+1 for a = m = 1..n, which is A003057 read as a triangle with offset 1. This entry was motivated by A249973 and A249974 by _Kerry Mitchell_ concerned with real values of r. - _Wolfdieter Lang_, Jan 11 2015
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Excircles.html">Excircles</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Exradius.html">Exradius</a>
%F T(n, m) = (n+1)(n-m+1), n >= m >= 1.
%F T(n, m) = rho_A = sqrt(s*(s-b)*(s-c)/(s-a)) with the semiperimeter s = (a + b + c)/2 and the substituted a, b, c values as given in the name. - _Wolfdieter Lang_, Dec 02 2014
%e The triangle T(n, m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e 1: 2
%e 2: 6 3
%e 3: 12 8 4
%e 4: 20 15 10 5
%e 5: 30 24 18 12 6
%e 6: 42 35 28 21 14 7
%e 7: 56 48 40 32 24 16 8
%e 8: 72 63 54 45 36 27 18 9
%e 9: 90 80 70 60 50 40 30 20 10
%e 10: 110 99 88 77 66 55 44 33 22 11
%e 11: 132 120 108 96 84 72 60 48 36 24 12
%e 12: 156 143 130 117 104 91 78 65 52 39 26 13
%e 13: 182 168 154 140 126 112 98 84 70 56 42 28 14
%e 14: 210 195 180 165 150 135 120 105 90 75 60 45 30 15
%e 15: 240 224 208 192 176 160 144 128 112 96 80 64 48 32 1
%e ... Formatted and extended by _Wolfdieter Lang_, Dec 02 2014
%e --------------------------------------------------------------
%e Example of general (a,b)-Fibonacci sequence positive integer limits r(a,b) (see the Jan 11 2015 comment above):
%e T(3, 2) = 8, that is a = m = 2 has a solution b = T(3, 2) = 8 with r = r(2,8) = n+1 = 4 = (2 + sqrt(4 + 4*8))/2. The other two solutions with r = 4 appear for b = T(3, m) with m = a = 1 and 3. In general, row n has n times the value n+1 for r, namely r(a=m,b=T(n,m)) = n+1, for m = 1..n. - _Wolfdieter Lang_, Jan 11 2015
%Y Cf. A003991 (incircle radius), A063930 (B-excircle radius), A001283 (C-excircle radius), A055096 (circumcircle diameter).
%K easy,nonn,tabl
%O 1,1
%A _Floor van Lamoen_, Aug 21 2001
%E Edited: Crossreferences commented and A055096 added by _Wolfdieter Lang_, Dec 02 2014
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