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A208855 Array of even catheti of primitive Pythagorean triangles when read by SW-NE diagonals. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..55.

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.

CROSSREFS

Cf. A208853, A208854.

Sequence in context: A258565 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

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Last modified September 25 21:33 EDT 2017. Contains 292500 sequences.