cp's OEIS Frontend

A063929 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.

%I A063929 #24 Feb 16 2025 08:32:45
%S A063929 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 A063929 24,16,8,72,63,54,45,36,27,18,9,90,80,70,60,50,40,30,20,10,110,99,88,
%U A063929 77,66,55,44,33,22,11,132,120,108,96,84,72,60,48,36,24,12,156,143,130,117
%N A063929 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 A063929 From _Wolfdieter Lang_, Dec 03 2014: (Start)
%C A063929 For excircles and their radii see the _Eric W. Weisstein_ links. Here the circle radius with center J_A is considered.
%C A063929 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 A063929 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 A063929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Excircles.html">Excircles</a>
%H A063929 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Exradius.html">Exradius</a>
%F A063929 T(n, m) = (n+1)(n-m+1), n >= m >= 1.
%F A063929 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 A063929 The triangle T(n, m) begins:
%e A063929 n\m   1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 ...
%e A063929 1:    2
%e A063929 2:    6   3
%e A063929 3:   12   8   4
%e A063929 4:   20  15  10  5
%e A063929 5:   30  24  18  12   6
%e A063929 6:   42  35  28  21  14   7
%e A063929 7:   56  48  40  32  24  16   8
%e A063929 8:   72  63  54  45  36  27  18   9
%e A063929 9:   90  80  70  60  50  40  30  20  10
%e A063929 10: 110  99  88  77  66  55  44  33  22 11
%e A063929 11: 132 120 108  96  84  72  60  48  36 24 12
%e A063929 12: 156 143 130 117 104  91  78  65  52 39 26 13
%e A063929 13: 182 168 154 140 126 112  98  84  70 56 42 28 14
%e A063929 14: 210 195 180 165 150 135 120 105  90 75 60 45 30 15
%e A063929 15: 240 224 208 192 176 160 144 128 112 96 80 64 48 32  1
%e A063929 ... Formatted and extended by _Wolfdieter Lang_, Dec 02 2014
%e A063929 --------------------------------------------------------------
%e A063929 Example of general (a,b)-Fibonacci sequence positive integer limits r(a,b) (see the Jan 11 2015 comment above):
%e A063929 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 A063929 Cf. A003991 (incircle radius), A063930 (B-excircle radius), A001283 (C-excircle radius), A055096 (circumcircle diameter).
%K A063929 easy,nonn,tabl
%O A063929 1,1
%A A063929 _Floor van Lamoen_, Aug 21 2001
%E A063929 Edited: Crossreferences commented and A055096 added by _Wolfdieter Lang_, Dec 02 2014