|
|
A349536
|
|
Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.
|
|
2
|
|
|
1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 75, 76, 77, 78, 79, 80, 84, 85, 86, 87, 89
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Number of Pythagorean triples with hypotenuse less than or equal to the next one.
|
|
REFERENCES
|
W. Sierpinski, Pythagorean Triangles, Dover Publications, 2003.
|
|
LINKS
|
|
|
FORMULA
|
Conjecture: the increment is a(n+1) - a(n) = 2^(m-1), where m is the sum of all powers of the Pythagorean primes (A002144) in the factorization of hypotenuse h(n+1) (see Eckert for PPT). However, starting from 58 the increment is 3.
|
|
EXAMPLE
|
The count of non-primitive Pythagorean triples as they appear in order of increasing hypotenuse:
.
Hypotenuse
-- ------------ --------------- ----
1 5 (3,4) 1
2 10 (6,8) 2
3 13 (5,12) 3
4 15 (9,12) 4
5 17 (8,15) 5
6 20 (12,16) 6
7 25 (7,24), (15,20) 8
8 26 (10,24) 9
9 29 (20,21) 10
|
|
PROG
|
(C)
// see enclosed main.c
for (long j=1; j< 101; ++j)
{
for (long k=1; k< 101; ++k)
{
if (k<=j) // to avoid pairs (as we need 1/8 or quarter plane)
{
double hyp=sqrt(j*j+k*k);
double c= (double) floor (hyp );
if (fabs(hyp - c) < DBL_EPSILON) arr[r++]= (long) c;
}}}
bubbleSort(arr, r); //sort by hypotenuse increase
for (long j=0; j< r; ++j)
{
if ( arr[j] != arr[j+1] )
{
// write to file: j is the sequence value a[n]*2
// arr[j] is the hypotenuse value
}
}
|
|
CROSSREFS
|
Cf. A349538 (extension to the full circle of Z^2 lattice).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|