A249866 Characteristic triangle for primitive Pythagorean triples.

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0

Offset

2

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.

T(n, m) = 1 if and only if there exists a primitive Pythagorean triple (x, y, z) with even y, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2.

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.

Links

Table of n, a(n) for n=2..98.

Formula

T(n, m) = 1 if n > m >= 1, (-1)^(n+m) = -1 and gcd(n, m) = 1, else 0.

Example

The triangle T(n, m) begins:
n\m  1  2  3  4  5  6  7  8  9 10 11 12 13  14 ...
--------------------------------------------------
2:   1
3:   0  1
4:   1  0  1
5:   0  1  0  1
6:   1  0  0  0  1
7:   0  1  0  1  0  1
8:   1  0  1  0  1  0  1
9:   0  1  0  1  0  0  0  1
10:  1  0  1  0  0  0  1  0  1
11:  0  1  0  1  0  1  0  1  0  1
12:  1  0  0  0  1  0  1  0  0  0  1
13:  0  1  0  1  0  1  0  1  0  1  0  1
14:  1  0  1  0  1  0  0  0  1  0  1  0  1
15:  0  1  0  1  0  0  0  1  0  0  0  0  0  1
...
T(5, 2) = 1 because the Pythagorean triple (21, 20, 29) is primitive (pairwise coprime).
T(5, 3) = 0 because the Pythagorean triple (16, 30, 34) is not primitive.

Prog

(PARI) {T(n, m) = n>m && m>0 && (n+m)%2 && gcd(n, m) ==1}; /* Michael Somos, Dec 05 2014 */

Crossrefs

Cf. A208853, A208854, A208855, A222946, A225949, A225950, A225952.

Sequence in context: A266719 A024711 A286990 * A369736 A128174 A096055

Adjacent sequences: A249863 A249864 A249865 * A249867 A249868 A249869

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Dec 03 2014

Status

approved