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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
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
Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Eric Weisstein's World of Mathematics, Pythagorean Triple
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
n (A009003(n)) Sides a(n)
-- ------------ --------------- ----
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).
Sequence in context: A327105 A356451 A129618 * A038673 A183219 A049533
KEYWORD
nonn,changed
AUTHOR
Alexander Kritov, Nov 21 2021
STATUS
approved