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%I #54 Oct 22 2021 07:06:07
%S 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,
%T 127,0,113,0,137,0,0,0,161,119,0,151,0,0,0,191,0,199,0,161,0,193,0,
%U 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
%N Triangle for sum of the two legs (catheti) of primitive Pythagorean triangles.
%C 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.
%C Here a(n,m) = 0 for non-primitive Pythagorean triangles.
%C 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).
%C The number of non-vanishing entries in row n is A055034(n).
%C The sequence of the main diagonal is 2*n^2-1 = A056220(n), n>= 2.
%C 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
%C 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)).
%C 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
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
%D 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.
%F 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.
%e The triangle a(n,m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 ...
%e 2: 7
%e 3: 0 17
%e 4: 23 0 31
%e 5: 0 41 0 49
%e 6: 47 0 0 0 71
%e 7: 0 73 0 89 0 97
%e 8: 79 0 103 0 119 0 127
%e 9: 0 113 0 137 0 0 0 161
%e 10: 119 0 151 0 0 0 191 0 199
%e 11: 0 161 0 193 0 217 0 233 0 241
%e 12: 167 0 0 0 239 0 263 0 0 0 287
%e ...
%e ---------------------------------------------------------
%e The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), with a(2,1) = 3 + 4 = 7.
%e The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), with a(7,4) = 33 + 56 = 89.
%e The primitive triangle for (n,m) = (8,1) is (x,y,z) = (63,16,65), with a(8,1) = 63 + 16 = 79.
%e 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.
%t T[n_, m_] := If[n > m >= 1 && GCD[n, m] == 1 && (-1)^(n+m) == -1, (n+m)^2 - 2 m^2, 0];
%t Table[T[n, m], {n, 2, 14}, {m, 1, n-1}] // Flatten (* _Jean-François Alcover_, Oct 22 2021 *)
%Y Cf. A222946 (hypotenuses), A222951 (perimeters), A056220 (main diagonals), A198441 (no zeros, ordered), A258149 (absolute leg differences), A278717 (leg differences).
%K nonn,easy,tabl
%O 2,1
%A _Wolfdieter Lang_, May 21 2013