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%I #15 Nov 30 2016 02:19:52
%S 1,0,7,7,0,17,0,1,0,31,23,0,0,0,49,0,17,0,23,0,71,47,0,7,0,41,0,97,0,
%T 41,0,7,0,0,0,127,79,0,31,0,0,0,89,0,161,0,73,0,17,0,47,0,119,0,199,
%U 119,0,0,0,1,0,73,0,0,0,241
%N Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.
%C For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
%C 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).
%C The number of non-vanishing entries in row n is A055034(n).
%C 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.
%C 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).
%C N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).
%C For the signed version 2*n*m - (n^2 - m^2) see A278717. - _Wolfdieter Lang_, Nov 30 2016
%D See also A225949.
%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.
%F 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.
%e The triangle a(n,m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 ...
%e 2: 1
%e 3: 0 7
%e 4: 7 0 17
%e 5: 0 1 0 31
%e 6: 23 0 0 0 49
%e 7: 0 17 0 23 0 71
%e 8: 47 0 7 0 41 0 97
%e 9: 0 41 0 7 0 0 0 127
%e 10: 79 0 31 0 0 0 89 0 161
%e 11: 0 73 0 17 0 47 0 119 0 199
%e 12: 119 0 0 0 1 0 73 0 0 0 241
%e ...
%e a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
%e a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
%e a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.
%t 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 *)
%Y Cf. A249866, A222946, A225949, A222951, A258150, A278717 (signed).
%K nonn,easy,tabl
%O 2,3
%A _Wolfdieter Lang_, Jun 10 2015
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