OFFSET
1,1
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).
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.
EXAMPLE
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
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 05 2012
STATUS
approved