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.

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

Alexander Kritov, Table of n, a(n) for n = 1..1050

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.

Alexander Kritov, C code that generates b-file

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. A008846, A009003.

Cf. A020882, A081804, A008846, A020883, A046086.

Cf. A349538 (extension to the full circle of Z^2 lattice).

Sequence in context: A327105 A356451 A129618 * A038673 A183219 A049533

Adjacent sequences: A349533 A349534 A349535 * A349537 A349538 A349539

Keyword

nonn

Author

Alexander Kritov, Nov 21 2021

Status

approved