A208855 Array of even catheti of primitive Pythagorean triangles when read by SW-NE diagonals.

4, 12, 8, 20, 24, 12, 28, 40, 0, 16, 36, 56, 60, 48, 20, 44, 72, 84, 80, 60, 24, 52, 88, 0, 112, 0, 0, 28, 60, 104, 132, 144, 140, 120, 84, 32, 68, 120, 156, 176, 180, 168, 140, 96, 36, 76, 136, 0, 208, 220, 0, 0, 160, 0, 40

Offset: 1

Author

Wolfdieter Lang, Mar 05 2012

Keywords

• nonn
• easy
• tabl

Comments

See the comments, reference and links in A208853. The present array is b(n,m) = 2*(2*n-1)*(2*m) if gcd(2*n-1,2*m)=1 and 0 otherwise. u=2*n-1, v=2*m. The array read by SW-NE diagonals is T(n,m):=b(n-m+1,m), n>=m>=1.

All primitive Pythagorean triples are given by

(a(n,m)=A208854(n,m),b(n,m),c(n,m)= A208853(n,m)), n>=1, m>=1. If the entry is 0 there is no primitive Pythagorean triple for these n and m values.

Examples

			Array b(n,m):
           m|  1    2    3    4    5    6    7    8    9   10 ...
           v|  2    4    6    8   10   12   14   16   18   20 ...
   n,  u
   1,  1       4    8   12   16   20   24   28   32   36   40 ...
   2,  3      12   24    0   48   60    0   84   96    0  120 ...
   3,  5      20   40   60   80    0  120  140  160  180    0 ...
   4,  7      28   56   84  112  140  168    0  224  252  280 ...
   5,  9      36   72    0  144  180    0  252  288    0  360 ...
   6, 11      44   88  132  176  220  264  308  352  396  440 ...
   7, 13      52  104  156  208  260  312  364  416  468  520 ...
   8, 15      60  120    0  240    0    0  420  480    0    0 ...
   9, 17      68  136  204  272  340  408  476  544  612  680 ...
  10, 19      76  152  228  304  380  456  532  608  684  760 ...
  ...
Triangle T(n,m):
           m|  1    2    3    4    5    6    7    8    9   10 ...
           v|  2    4    6    8   10   12   14   16   18   20 ...
   n,  u
   1,  1       4
   2,  3      12    8
   3,  5      20   24   12
   4,  7      28   40    0   16
   5,  9      36   56   60   48   20
   6, 11      44   72   84   80   60   24
   7, 13      52   88    0  112    0    0   28
   8, 15      60  104  132  144  140  120   84   32
   9, 17      68  120  156  176  180  168  140   96   36
  10, 19      76  136    0  208  220    0    0  160    0   40
  ...
For some triples see the example section of A208853.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Crossrefs

Cf. A208853, A208854.

Programs

Mathematica

A208855[n_, m_] := If[CoprimeQ[#, 2*m], 4*m*#, 0] & [2*(n-m) + 1];
Table[A208855[n, m], {n, 15}, {m, n}] (* Paolo Xausa, Feb 12 2025 *)

Formula

T(n,m)=b(n-m+1,m), n>=m>=1, with b(n,m) = 4*(2*n-1)*m if gcd(2*n-1,2*m)=1 and 0 otherwise.