|
%I #12 Mar 30 2012 18:49:34
%S 4,12,8,20,24,12,28,40,0,16,36,56,60,48,20,44,72,84,80,60,24,52,88,0,
%T 112,0,0,28,60,104,132,144,140,120,84,32,68,120,156,176,180,168,140,
%U 96,36,76,136,0,208,220,0,0,160,0,40
%N Array of even catheti of primitive Pythagorean triangles when read by SW-NE diagonals.
%C 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.
%C All primitive Pythagorean triples are given by
%C (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.
%F 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.
%e Array b(n,m):
%e ........m| 1 2 3 4 5 6 7 8 9 10 ...
%e ........v| 2 4 6 8 10 12 14 16 18 20
%e n, u
%e 1, 1 4 8 12 16 20 24 28 32 36 40
%e 2, 3 12 24 0 48 60 0 84 96 0 120
%e 3, 5 20 40 60 80 0 120 140 160 180 0
%e 4, 7 28 56 84 112 140 168 0 224 252 280
%e 5, 9 36 72 0 144 180 0 252 288 0 360
%e 6, 11 44 88 132 176 220 264 308 352 396 440
%e 7, 13 52 104 156 208 260 312 364 416 468 520
%e 8, 15 60 120 0 240 0 0 420 480 0 0
%e 9, 17 68 136 204 272 340 408 476 544 612 680
%e 10,19 76 152 228 304 380 456 532 608 684 760
%e ...
%e Triangle T(n,m):
%e ......m| 1 2 3 4 5 6 7 8 9 10
%e ......v| 2 4 6 8 10 12 14 16 18 20
%e n, u
%e 1, 1 4
%e 2, 3 12 8
%e 3, 5 20 24 12
%e 4, 7 28 40 0 16
%e 5, 9 36 56 60 48 20
%e 6, 11 44 72 84 80 60 24
%e 7, 13 52 88 0 112 0 0 28
%e 8, 15 60 104 132 144 140 120 84 32
%e 9, 17 68 120 156 176 180 168 140 96 36
%e 10,19 76 136 0 208 220 0 0 160 0 40
%e ...
%e For some triples see the example section of A208853.
%Y Cf. A208853, A208854.
%K nonn,easy,tabl
%O 1,1
%A _Wolfdieter Lang_, Mar 05 2012
|
