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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A225950 Triangle for odd legs of primitive Pythagorean triangles.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, May 23 2013

Keywords

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

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.

A120891 Number of primitive Pythagorean triangles with odd leg 2n-1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 4, 1, 1, 2, 2
Offset: 1

Views

Author

Lekraj Beedassy, Jul 12 2006

Keywords

Comments

Bisection of A024361.
Bisection of even-numbered terms of A024361 results in alternating zero terms; removing zeros gives A068068. - Ray Chandler, Feb 04 2020

Links

Crossrefs

Cf. A024361, A068068, A092523, A120890.

Formula

a(n)=2^(k-1), where k=A092523(n) for n > 1.

Extensions

a(1)=0 inserted by Ray Chandler, Feb 04 2020

A354570 Ordered odd leg lengths k (listed with multiplicity) of primitive Pythagorean triangles such that all prime factors of k are congruent to 3 (mod 4).

Original entry on oeis.org

3, 7, 9, 11, 19, 21, 21, 23, 27, 31, 33, 33, 43, 47, 49, 57, 57, 59, 63, 63, 67, 69, 69, 71, 77, 77, 79, 81, 83, 93, 93, 99, 99, 103, 107, 121, 127, 129, 129, 131, 133, 133, 139, 141, 141, 147, 147, 151, 161, 161, 163, 167, 171, 171, 177, 177, 179, 189, 189, 191
Offset: 1

Views

Author

Lothar Selle, Jun 03 2022

Keywords

Comments

Conjecture: lim_{n->oo} a(n)/n = Pi. Also, lim_{n->oo} A354571(n)/n = Pi.

Examples

			3 is a term: 3^2 + 4^2 = 5^2, so the triangle with sides (3,4,5) is a Pythagorean triangle; GCD(3,4,5) = 1, so it is primitive; and the odd leg length, 3, has no prime factors p that are not congruent to 3 (mod 4).
5 is not a term: it is the odd leg length of the primitive Pythagorean triangle (5,12,13), but 5 (a prime) == 1 (mod 4).
21 (whose prime factors are 3 and 7, both of which are congruent to 3 (mod 4)) is listed twice because it is the odd leg length of two primitive Pythagorean triangles ((20,21,29) and (21,220,221)).

References

  • Lothar Selle, Kleines Handbuch Pythagoreische Zahlentripel, Books on Demand, 4th impression 2022, chapter 2.2.1; see chapter 2.3.10 for identity of lim_(n->oo) A354571(n)/n.

Crossrefs

Intersection of A004614 and A120890.
Cf. A354571.
Showing 1-3 of 3 results.

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