cp's OEIS Frontend

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.

%I A349536 #52 Feb 16 2025 08:34:02
%S A349536 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,
%T A349536 31,32,33,34,35,37,38,39,40,44,45,46,47,48,49,50,52,53,54,55,56,57,58,
%U A349536 59,60,61,62,63,64,65,66,67,68,71,75,76,77,78,79,80,84,85,86,87,89
%N 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.
%C A349536 Number of Pythagorean triples with hypotenuse less than or equal to the next one.
%D A349536 W. Sierpinski, Pythagorean Triangles, Dover Publications, 2003.
%H A349536 Alexander Kritov, <a href="/A349536/b349536.txt">Table of n, a(n) for n = 1..1050</a>
%H A349536 Manuel Benito and Juan L. Varona, <a href="https://doi.org/10.1016/S0377-0427(01)00496-4">Pythagorean triangles with legs less than n</a>, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
%H A349536 E. Eckert, <a href="http://www.jstor.org/stable/2690291">The group of primitive Pythagorean triangles</a>, Mathematics Magazine 57 (1984) 22-27.
%H A349536 Alexander Kritov, <a href="/A349536/a349536.c.txt">C code that generates b-file</a>
%H A349536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A349536 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.
%e A349536 The count of non-primitive Pythagorean triples as they appear in order of increasing hypotenuse:
%e A349536 .
%e A349536        Hypotenuse
%e A349536    n  (A009003(n))       Sides       a(n)
%e A349536   --  ------------  ---------------  ----
%e A349536    1        5            (3,4)         1
%e A349536    2       10            (6,8)         2
%e A349536    3       13            (5,12)        3
%e A349536    4       15            (9,12)        4
%e A349536    5       17            (8,15)        5
%e A349536    6       20           (12,16)        6
%e A349536    7       25       (7,24), (15,20)    8
%e A349536    8       26           (10,24)        9
%e A349536    9       29           (20,21)       10
%o A349536 (C)
%o A349536 // see enclosed main.c
%o A349536 for (long j=1;j< 101;++j)
%o A349536 {
%o A349536 for (long k=1;k< 101;++k)
%o A349536 {
%o A349536 if (k<=j)   // to avoid pairs (as we need 1/8 or quarter plane)
%o A349536     {
%o A349536           double hyp=sqrt(j*j+k*k);
%o A349536           double c= (double) floor (hyp );
%o A349536 if   (fabs(hyp - c) < DBL_EPSILON)  arr[r++]= (long) c;
%o A349536 }}}
%o A349536 bubbleSort(arr, r);//sort by hypotenuse increase
%o A349536 for (long j=0;j< r;++j)
%o A349536 {
%o A349536    if  ( arr[j] != arr[j+1] )
%o A349536     {
%o A349536         // write to file: j is the sequence value a[n]*2
%o A349536         // arr[j] is the hypotenuse value
%o A349536     }
%o A349536 }
%Y A349536 Cf. A008846, A009003.
%Y A349536 Cf. A020882, A081804, A008846, A020883, A046086.
%Y A349536 Cf. A349538 (extension to the full circle of Z^2 lattice).
%K A349536 nonn
%O A349536 1,2
%A A349536 _Alexander Kritov_, Nov 21 2021