A225950 Triangle for odd legs of primitive Pythagorean triangles.

3, 0, 5, 15, 0, 7, 0, 21, 0, 9, 35, 0, 0, 0, 11, 0, 45, 0, 33, 0, 13, 63, 0, 55, 0, 39, 0, 15, 0, 77, 0, 65, 0, 0, 0, 17, 99, 0, 91, 0, 0, 0, 51, 0, 19, 0, 117, 0, 105, 0, 85, 0, 57, 0, 21, 143, 0, 0, 0, 119, 0, 95, 0, 0, 0, 23, 0, 165, 0, 153, 0, 133, 0, 105, 0, 69, 0, 25, 195, 0, 187, 0, 171, 0, 0, 0, 115, 0, 75, 0, 27, 0, 221, 0, 209, 0, 0, 0, 161, 0, 0, 0, 0, 0, 29

Offset: 2

Author

Wolfdieter Lang, May 23 2013

Keywords

• nonn
• easy
• tabl

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 mirror triangles with x even are not considered here. Therefore a(n,m) = n^2 - m^2 (for these solutions).

The number of non-vanishing entries in row n is A055034(n).

The sequence of the main diagonal is 2*n -1 = A005408(n-1),

n >= 2.

If the zeros are eliminated and the numbers are sorted nondecreasingly (multiple entries appear) one obtains A120890. All odd numbers >= 3 appear, they are given in A005408. Note that all odd legs x will be found if one takes in the triangle n = 2, ..., floor((x+1)/2).

Examples

			The triangle a(n,m) begins:
n\m   1    2   3    4    5    6   7     8   9   10  11  12 ...
2:    3
3:    0    5
4:   15    0   7
5:    0   21   0    9
6:   35    0   0    0   11
7:    0   45   0   33    0   13
8:   63    0  55    0   39    0  15
9:    0   77   0   65    0    0   0    17
10:  99    0  91    0    0    0  51     0  19
11:   0  117   0  105    0   85   0    57   0   21
12: 143    0   0    0  119    0  95     0   0    0  23
13:   0  165   0  153    0  133   0   105   0   69   0  25
...
a(6,1) = 35 from the primitive triangle (35,12,37).
a(6,2) = 0 because n and m are even (not allowed n, m values for primitive triangles).
a(6,3) = 0 because gcd(6,3) = 3 (not 1, hence not allowed).
		

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

• Kival Ngaokrajang, Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..50.

Crossrefs

Cf. A222946 (hypotenuses), A225952 (even legs), A225949 (leg sums), A225951 (perimeters), A120890 (odd legs, ordered).

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.