OFFSET
2,1
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.
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,k) = A055096(n-1,k) * ((n+k) mod 2) * A063524 (gcd(n,k)): terms in A055096 that are not hypotenuses in primitive Pythagorean triangles, are replaced by 0. - Reinhard Zumkeller, Mar 23 2013
The number of non-vanishing entries in row n is A055034(n). - Wolfdieter Lang, Mar 24 2013
The non-vanishing entries when ordered according to nondecreasing leg sums x+y (see A225949 and A198441) produce (with multiplicities) A198440. - Wolfdieter Lang, May 22 2013
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
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
Reinhard Zumkeller, Rows n = 2..120 of triangle, flattened
FORMULA
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.
EXAMPLE
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
...
------------------------------------------------------------
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).
PROG
(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..]
-- Reinhard Zumkeller, Mar 23 2013
CROSSREFS
AUTHOR
Wolfdieter Lang, Mar 21 2013
STATUS
approved