A014498 and JHC's extraversion

[Antreas P. Hatzipolakis]

ID Number: A014498
Sequence:  1,2,3,3,6,5,4,10,5,12,7,15,14,6,15,20,9,21,7,18,28,11,8,21,
           30,35,22,9,36,24,35,13,42,33,45,10,26,40,44,15,39,11,30,45,
           55,56,30,63,52,12,33,66,17,63,65,72
Name:      Varying radii of inscribed circles within primitive Pythagorean
           triples as a function of increasing values of hypotenuse.

___________________________________________________________________________

The sequence above(*) is inradius-related.
(*) Note for the Hyacinthians: It is from Sloane's _Encycl.of Int. Sequences_
        http://www.research.att.com/~njas/sequences/index.html

Let's see if JHC's "extraversion"(**) applies here for the three exradii .
(**): Note for the seqfans: If you want to learn about this and other things
on triangle geometry, join our group Hyacinthos !

Let (z = m^2 + n^2: hypot., (y = 2mn, x = m^2 - n^2): legs) be a pythagorean
triangle.

                  area(ABC)        xy            2mn(m^2-n^2)
r [=inradius] = -------------- =  ---- = ------------------------------  =
                 semiperimeter     2s      m^2 + n^2 + 2mn + m^2 - n^2

= n(m-n)
                      area(ABC)     xy           2mn(m^2-n^2)
r_z [= z-exradius] = ---------- = ----- = ------------------------------ =
                        s-z       2(s-z)  -(m^2 + n^2) + 2mn + m^2 - n^2

= m(m+n)

                      area(ABC)     xy           2mn(m^2-n^2)
r_y [= y-exradius] = ---------- = ----- = ------------------------------ =
                        s-y       2(s-y)    m^2 + n^2 - 2mn + m^2 - n^2

= n(m+n)

                      area(ABC)     xy           2mn(m^2-n^2)
r_x [= x-exradius] = ---------- = ----- = ------------------------------- =
                        s-x       2(s-x)    m^2 + n^2 + 2mn - (m^2 - n^2)

= m(m-n)

So, in a pythagorean triangle all 4 in/exradii are integral.

Let's now denote: c = z (hypotenuse)
                  b = max(y,x) (middle side)
                  a = min(y,x) (smallest side)
and r, r_c, r_b, r_a the corresponding in/exradii.

                              TABLE

              A046087
        A020882  | A046086   A014498
m    n     c     b    a         r       r_c    r_b     r_a
-------------------------------------------------------------------------

2    1     5     4    3         1        6      3       2
3    2    13    12    5         2       15     10       3
4    1    17    15    8         3       20     12       5
4    3    25    24    7         3       28     21       4
5    2    29    21   20         6       35     15      14
6    1    37    35   12         5       42     30       7
5    4    41    40    9         4       45     36       5
7    2    53    45   28        10       63     35      18
6    5    61    60   11         5       66     55       5
7    4    65    56   33        12       77     44      21
8    1    65    63   16         7       72     56       9
8    3    73    55   48        15       88     40      33
9    2    85    77   36        14       99     22      14
7    6    85    84   13         6       91     78       7
8    5    89    80   39        15      104     65      24
9    4    97    72   65        20      117     52      45
..........

Sequences for the three exradii:

1. r_c sequence: 6,15,20,28,35,42,45,63,66,77,72,88,99,91,104,117,....
Name:      Varying radii of c-exscribed circles within primitive Pythagorean
           triples as a function of increasing values of hypotenuse c.
Formula:   Arrange all primitive Pythagorean triples a,b,c by value of
           hypotenuse c; for nth value of c, sequence gives radius of
           c-exscribed circle, (a+b+c)/2.
           a(n) = (A046086(n) + A046087(n) + A020882(n))/2
Note: This sequence ordered gives:
ID Number: A020886
Sequence:  6,15,20,28,35,42,45,63,66,72,77,88,91,99,104,110,117,120,130,143,
153,156,165,170,187,190,195,204,209,210,221,228,231,238,247,255,266,272,273,276
Name:      Ordered semiperimeters of primitive Pythagorean triangles.


2. r_b sequence: 3,10,12,21,15,30,36,35,55,44,56,40,22,78,65,52,.....
Name:      Varying radii of b-exscribed circles within primitive Pythagorean
           triples as a function of increasing values of hypotenuse c, and
           b as middle side.


3. r_c sequence: 2,3,5,4,14,7,5,18,5,21,9,33,14,7,24,45,.....
Name:      Varying radii of a-exscribed circles within primitive Pythagorean
           triples as a function of increasing values of hypotenuse c, and
           a as smallest side.

.
Antreas