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A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles. 3
1, 0, 7, 7, 0, 17, 0, 1, 0, 31, 23, 0, 0, 0, 49, 0, 17, 0, 23, 0, 71, 47, 0, 7, 0, 41, 0, 97, 0, 41, 0, 7, 0, 0, 0, 127, 79, 0, 31, 0, 0, 0, 89, 0, 161, 0, 73, 0, 17, 0, 47, 0, 119, 0, 199, 119, 0, 0, 0, 1, 0, 73, 0, 0, 0, 241 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.

Here a(n,m) = 0 for non-primitive Pythagorean triangles, and for primitive Pythagorean triangles a(n,m) = abs(n^2 - m^2 - 2*n*m) = abs((n-m)^2 - 2*m^2).

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

D(n,m):= n^2 - m^2 - 2*n*m >= 0 if 1 <= m <= floor(n/(sqrt(2)+1)), and D(n,m) < 0 if n/(sqrt(2)+1)+1 <= m <= n-1, for n >= 2.

The Pell equation (n-m)^2 - 2*m^2 = +/- N is important here to find the representations of +N or -N in the triangle D(n,m). For instance, odd primes N have to be of the +1 (mod 8) (A007519) or -1 (mod 8) (A007522) form, that is, from A001132. See the Nagell reference, Theorem 110, p. 208 with Theorem 111, pp. 210-211. E.g., N = +7 appears for m = 1, 3, 9, 19, 53, ... (A077442) for n = 4, 8, 22, 46, 128, ... (2*A006452).

  N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).

For the  signed version 2*n*m - (n^2 - m^2) see A278717. - Wolfdieter Lang, Nov 30 2016

REFERENCES

See also A225949.

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.

LINKS

Table of n, a(n) for n=2..67.

FORMULA

a(n,m) =  abs(n^2 - m^2 -2*n*m) = abs((n-m)^2 - 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 ...

2:    1

3:    0  7

4:    7  0 17

5:    0  1  0 31

6:   23  0  0  0 49

7:    0 17  0 23  0 71

8:   47  0  7  0 41  0 97

9:    0 41  0  7  0  0  0 127

10:  79  0 31  0  0  0 89   0 161

11:   0 73  0 17  0 47  0 119   0 199

12: 119  0  0  0  1  0 73   0   0   0 241

...

a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.

a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.

a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.

MATHEMATICA

a[n_, m_] /; n > m >= 1 && CoprimeQ[n, m] && (-1)^(n+m) == -1 := Abs[n^2 - m^2 - 2*n*m]; a[_, _] = 0; Table[a[n, m], {n, 2, 12}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jun 16 2015, after given formula *)

CROSSREFS

Cf. A249866, A222946, A225949, A222951, A258150, A278717 (signed).

Sequence in context: A064890 A196602 A200622 * A278717 A241009 A278657

Adjacent sequences:  A258146 A258147 A258148 * A258150 A258151 A258152

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jun 10 2015

STATUS

approved

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Last modified September 15 20:35 EDT 2019. Contains 327087 sequences. (Running on oeis4.)