A208854 Array of odd catheti of primitive Pythagorean triangles when read by SW-NE diagonals.

3, 5, 15, 21, 7, 35, 45, 9, 0, 63, 77, 33, 11, 55, 99, 117, 65, 13, 39, 91, 143, 165, 105, 0, 15, 0, 0, 195, 221, 153, 85, 17, 51, 119, 187, 255, 285, 209, 133, 57, 19, 95, 171, 247, 323, 357, 273, 0, 105, 21, 0, 0, 231, 0, 399

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 a(n,m) = abs((2*n-1)^2 - (2*m)^2) if gcd(2*n-1,2*m)=1 and 0 otherwise. Put u=2*n-1 and v=2*m. The array read by SW-NE diagonals is T(n,m):=a(n-m+1,m), n>=m>=1.

All primitive Pythagorean triples are given by (a(n,m),b(n,m):=A208855(n,m), c(n,m):=A208853(n,m)), n>=1, m>=1. If the entry is (0,0,0) there is no primitive Pythagorean triple for these n and m values. See the example section of A208853 for the array of triples.

Every odd number a=2*k+1, k>=1, appears at least in one primitive triple, namely in (2*k+1, 4*T(k),4*T(k)+1), with the triangular numbers T(k) := A000217(k). This a-value is a=u^2-v^2 with (u,v)=(k+1,k). It may appear in other primitive triples. E.g. a=33=2*16+1 appears in (u,v)=(17,16) ((n,m)= (9,8)) as (33,544,545), and also in (33,56,65) with (n,m)=(4,2) (maybe others).

Examples

			Array a(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     3   15   35    63    99   143   195    255   323    399
2,  3     5    7    0    55    91     0   187    247     0    391
3,  5    21    9   11    39     0   119   171    231   299      0
4,  7    45   33   13    15    51    95     0    207   275    351
5,  9    77   65    0    17    19     0   115    175     0    319
6, 11   117  105   85    57    21    23    75    135   203    279
7, 13   165  153  133   105    69    25    27     87   155    231
8, 15   221  209    0   161     0     0    29     31     0      0
9, 17   285  273  253   225   189   145    93     33    35    111
10,19   357  345  325   297   261   217   165    105    37     39
...
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       3
2,  3       5    15
3,  5      21     7    35
4,  7      45     9     0     63
5,  9      77    33    11     55    99
6  11     117    65    13     39    91   143
7, 13     165   105     0     15     0     0    195
8, 15     221   153    85     17    51   119    187   255
9, 17     285   209   133     57    19    95    171   247   323
10,19     357   273     0    105    21     0      0   231     0  399
...
For the array of triples see the example section of A208853.
		

Crossrefs

Cf. A208853, A208855.

Formula

T(n,m)=a(n-m+1,m), n>=m>=1, with a(n,m):=abs((2*n-1)^2 - (2*m)^2) if gcd(2*n-1,2*m)=1 and 0 otherwise.