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A208855
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Array of even catheti of primitive Pythagorean triangles when read by SW-NE diagonals.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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