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
Paolo Xausa, Table of n, a(n) for n = 2..11176 (rows 2..150 of triangle, flattened).
Wolfdieter Lang, On Positive Integer Descartes-Steiner Curvature Quintuplets, arXiv:2503.08631 [math.NT], 2025. See p. 8.
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.
MATHEMATICA
A249866[n_, m_] := Boole[OddQ[n + m] && CoprimeQ[n, m]];
Table[A249866[n, m], {n, 2, 15}, {m, n - 1}] (* Paolo Xausa, Feb 12 2025 *)
PROG
(PARI) {T(n, m) = n>m && m>0 && (n+m)%2 && gcd(n, m) ==1}; /* Michael Somos, Dec 05 2014 */
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 03 2014
STATUS
approved