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
Keywords
Examples
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
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
Crossrefs
Programs
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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 } }
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.
Comments