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

1,1

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.

Links

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

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.

Example

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.

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 *)

Crossrefs

Cf. A208853, A208854.

Sequence in context: A310569 A145046 A205849 * A252984 A084415 A156681

Adjacent sequences: A208852 A208853 A208854 * A208856 A208857 A208858

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 05 2012

Status

approved