OFFSET
2,3
COMMENTS
For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
Here a(n,m) = 0 for non-primitive Pythagorean triangles, and for primitive Pythagorean triangles a(n,m) = abs(n^2 - m^2 - 2*n*m) = abs((n-m)^2 - 2*m^2).
The number of non-vanishing entries in row n is A055034(n).
D(n,m):= n^2 - m^2 - 2*n*m >= 0 if 1 <= m <= floor(n/(sqrt(2)+1)), and D(n,m) < 0 if n/(sqrt(2)+1)+1 <= m <= n-1, for n >= 2.
The Pell equation (n-m)^2 - 2*m^2 = +/- N is important here to find the representations of +N or -N in the triangle D(n,m). For instance, odd primes N have to be of the +1 (mod 8) (A007519) or -1 (mod 8) (A007522) form, that is, from A001132. See the Nagell reference, Theorem 110, p. 208 with Theorem 111, pp. 210-211. E.g., N = +7 appears for m = 1, 3, 9, 19, 53, ... (A077442) for n = 4, 8, 22, 46, 128, ... (2*A006452).
For the signed version 2*n*m - (n^2 - m^2) see A278717. - Wolfdieter Lang, Nov 30 2016
REFERENCES
See also A225949.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.
FORMULA
a(n,m) = abs(n^2 - m^2 -2*n*m) = abs((n-m)^2 - 2*m^2) if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^(n+m) = -1); otherwise a(n,m) = 0.
EXAMPLE
The triangle a(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 ...
2: 1
3: 0 7
4: 7 0 17
5: 0 1 0 31
6: 23 0 0 0 49
7: 0 17 0 23 0 71
8: 47 0 7 0 41 0 97
9: 0 41 0 7 0 0 0 127
10: 79 0 31 0 0 0 89 0 161
11: 0 73 0 17 0 47 0 119 0 199
12: 119 0 0 0 1 0 73 0 0 0 241
...
a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.
MATHEMATICA
a[n_, m_] /; n > m >= 1 && CoprimeQ[n, m] && (-1)^(n+m) == -1 := Abs[n^2 - m^2 - 2*n*m]; a[_, _] = 0; Table[a[n, m], {n, 2, 12}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jun 16 2015, after given formula *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 10 2015
STATUS
approved