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A222946 Triangle for hypotenuses of primitive Pythagorean triangles. 17
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 (list; table; graph; refs; listen; history; text; internal format)
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) gives also 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. - 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

Cf. A020882 (ordered nonzero values a(n,m) with multiplicity).

Cf. A249866, A225950 (odd legs), A225951 (perimeters), A225952 (even legs), A225949 (leg sums), A249869 (areas), A258149 (absolute leg differences), A278717 (leg differences).

Sequence in context: A007392 A292105 A052401 * A214121 A024418 A167297

Adjacent sequences:  A222943 A222944 A222945 * A222947 A222948 A222949

KEYWORD

nonn,easy,tabl,look

AUTHOR

Wolfdieter Lang, Mar 21 2013

STATUS

approved

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)