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

2,1

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 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).

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

Table of n, a(n) for n=2..106.

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

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 ...
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).

Crossrefs

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

Sequence in context: A088521 A022837 A328631 * A344503 A088969 A009780

Adjacent sequences: A225947 A225948 A225949 * A225951 A225952 A225953

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, May 23 2013

Status

approved