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A222946
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Triangle for hypotenuses of primitive Pythagorean triangles.
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17
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5, 0, 13, 17, 0, 25, 0, 29, 0, 41, 37, 0, 0, 0, 61, 0, 53, 0, 65, 0, 85, 65, 0, 73, 0, 89, 0, 113, 0, 85, 0, 97, 0, 0, 0, 145, 101, 0, 109, 0, 0, 0, 149, 0, 181, 0, 125, 0, 137, 0, 157, 0, 185, 0, 221, 145, 0, 0, 0, 169, 0, 193, 0, 0, 0, 265, 0, 173, 0, 185, 0, 205, 0, 233, 0, 269, 0, 313, 197, 0, 205, 0, 221, 0, 0, 0, 277, 0, 317, 0, 365
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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 diagonal sequence is given by a(n,n-1) = A001844(n-1), n >= 2.
The row sums of this triangle are 5, 13, 42, 70, 98, 203, 340, 327, 540, ...
a(n, m) also gives twice the member s(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*t(n, m) = A225949(n, m). See A278717 for details and the Keith Conrad reference there. - 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^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 12 13 ...
2: 5
3: 0 13
4: 17 0 25
5: 0 29 0 41
6: 37 0 0 0 61
7: 0 53 0 65 0 85
8: 65 0 73 0 89 0 113
9: 0 85 0 97 0 0 0 145
10: 101 0 109 0 0 0 149 0 181
11: 0 125 0 137 0 157 0 185 0 221
12: 145 0 0 0 169 0 193 0 0 0 265
13: 0 173 0 185 0 205 0 233 0 269 0 313
14: 197 0 205 0 221 0 0 0 277 0 317 0 365
...
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a(7,4) = 7^2 + 4^2 = 49 + 16 = 65.
a(8,1) = 8^2 + 1^2 = 64 + 1 = 65.
a(3,1) = 0 because n and m are both odd.
a(4,2) = 0 because n and m are both even.
a(6,3) = 0 because gcd(6,3) = 3 (not 1).
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5).
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65).
The primitive triangle for (n,m) = (8,1) is (x,y,z) = (63,16,65).
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PROG
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(Haskell)
a222946 n k = a222946_tabl !! (n-2) !! (k-1)
a222946_row n = a222946_tabl !! (n-2)
a222946_tabl = zipWith p [2..] a055096_tabl where
p x row = zipWith (*) row $
map (\k -> ((x + k) `mod` 2) * a063524 (gcd x k)) [1..]
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CROSSREFS
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Cf. A020882 (ordered nonzero values a(n,m) with multiplicity).
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KEYWORD
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AUTHOR
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STATUS
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approved
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