Search: a222951 -id:a222951
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A225949
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Triangle for sum of the two legs (catheti) of primitive Pythagorean triangles.
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+10
10
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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
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OFFSET
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2,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
...
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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.
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MATHEMATICA
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T[n_, m_] := If[n > m >= 1 && GCD[n, m] == 1 && (-1)^(n+m) == -1, (n+m)^2 - 2 m^2, 0];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A258149
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Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.
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+10
3
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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, 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, 119, 0, 0, 0, 1, 0, 73, 0, 0, 0, 241
(list;
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history;
text;
internal format)
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OFFSET
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2,3
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COMMENTS
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For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
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).
The number of non-vanishing entries in row n is A055034(n).
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.
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).
N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).
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REFERENCES
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T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.
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LINKS
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FORMULA
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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.
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EXAMPLE
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The triangle a(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11 ...
2: 1
3: 0 7
4: 7 0 17
5: 0 1 0 31
6: 23 0 0 0 49
7: 0 17 0 23 0 71
8: 47 0 7 0 41 0 97
9: 0 41 0 7 0 0 0 127
10: 79 0 31 0 0 0 89 0 161
11: 0 73 0 17 0 47 0 119 0 199
12: 119 0 0 0 1 0 73 0 0 0 241
...
a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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