A225949 Triangle for sum of the two legs (catheti) of primitive Pythagorean triangles.

7, 0, 17, 23, 0, 31, 0, 41, 0, 49, 47, 0, 0, 0, 71, 0, 73, 0, 89, 0, 97, 79, 0, 103, 0, 119, 0, 127, 0, 113, 0, 137, 0, 0, 0, 161, 119, 0, 151, 0, 0, 0, 191, 0, 199, 0, 161, 0, 193, 0, 217, 0, 233, 0, 241, 167, 0, 0, 0, 239, 0, 263, 0, 0, 0, 287, 0, 217, 0, 257, 0, 289, 0, 313, 0, 329, 0, 337, 223, 0, 271, 0, 311, 0, 0, 0, 367, 0, 383, 0, 391

Offset: 2

Author

Wolfdieter Lang, May 21 2013

Keywords

• nonn
• easy
• tabl

Comments

For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p. 190.

Here a(n,m) = 0 for non-primitive Pythagorean triangles.

There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2. The mirror triangles with x even are not considered here. Therefore a(n,m) = n^2 - m^2 + 2*n*m (for these solutions).

The number of non-vanishing entries in row n is A055034(n).

The sequence of the main diagonal is 2*n^2-1 = A056220(n), n>= 2.

The sequence of the main diagonal is j^2 + k^2 - 2 or 2*j*k if n>=2 and j = n + sqrt(2)/2 and k = n - sqrt(2)/2. - Avi Friedlich, Mar 30 2015

If the 0 entries are eliminated and the numbers are ordered increasingly (keeping multiple entries) the sequence becomes A198441(n-1), n>=2. If multiple entries are recorded only once this becomes A058529 (a proper subsequence of A118905). Note that all leg sums <= N are certainly reached if one considers rows n = 2, ..., floor(-1 + sqrt(N+2)).

a(n, m) also gives twice the member t(n, m) of the triple (r(n, m), s(n, m), t(n, m)) with squares r(n, m)^2, s(n, m)^2 and t(n, m)^2 in arithmetic progression with common difference A(n, m) = A249869(n, m), the area of the primitive Pythagorean triangle, or 0 if there is no such triangle. The other members are given by 2*r(n, m) = A278717(n, m) and 2*s(n, m) = A222946(n, m). See A278717 for details and the Keith Conrad reference. - Wolfdieter Lang, Nov 30 2016

Examples

			The triangle a(n,m) begins:
n\m   1   2   3   4   5   6    7    8    9   10   11 ...
2:    7
3:    0  17
4:   23   0  31
5:    0  41   0  49
6:   47   0   0   0  71
7:    0  73   0  89   0  97
8:   79   0 103   0 119   0  127
9:    0 113   0 137   0   0    0  161
10: 119   0 151   0   0   0  191    0  199
11:   0 161   0 193   0 217    0  233    0  241
12: 167   0   0   0 239   0  263    0    0    0  287
...
---------------------------------------------------------
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), with a(2,1) = 3 + 4 = 7.
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), with a(7,4) = 33 + 56 = 89.
The primitive triangle for (n,m) = (8,1) is (x,y,z) = (63,16,65), with a(8,1) = 63 + 16 = 79.
All primitive Pythagorean triangles with leg sums <= 167 are certainly covered by this triangle (rows n = 2..12), and the multiplicities are also correct, e.g., 119 appears twice.
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
• Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

Crossrefs

Cf. A222946 (hypotenuses), A222951 (perimeters), A056220 (main diagonals), A198441 (no zeros, ordered), A258149 (absolute leg differences), A278717 (leg differences).

Programs

Mathematica

T[n_, m_] := If[n > m >= 1 && GCD[n, m] == 1 && (-1)^(n+m) == -1, (n+m)^2 - 2 m^2, 0];
Table[T[n, m], {n, 2, 14}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Oct 22 2021 *)

Formula

a(n,m) = (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.