A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

1, 5, 9, 13, 17, 29, 33, 37, 41, 45, 57, 61, 65, 77, 81, 93, 97, 109, 113, 117, 129, 133, 137, 141, 145, 165, 177, 181, 185, 197, 209, 213, 217, 221, 233, 245, 249, 261, 265, 277, 289, 301, 305, 309, 313, 325, 329, 333, 337, 341, 361, 373, 385, 397, 401, 413, 417, 421, 433, 437, 449

Offset: 0

Author

Alexander Kritov, Nov 21 2021

Keywords

• nonn

Comments

Consider a 2D lattice, where the Cartesian coordinates x and y are legs of the Pythagorean triangle. Thus the notion of Pythagorean triple is extended to the cases when sides x, y are in Z (i.e., sides also include negative integers and zero). The sequence gives the number of such triples on or inside a circle of radius n.

Partial sums of A046109.

Examples

			Sides (coordinates)                                                       a(n)
------------------------------------------------------------------------------
(0,0)                                                                       1
(-1,0)(0,-1)(0,1)(1,0)                                                      5
(-2,0)(0,-2)(0,2)(2,0)                                                      9
(-3,0)(0,-3)(0,3)(3,0)                                                     13
(-4,0)(0,-4)(0,4)(4,0)                                                     17
(-5,0)(-4,-3)(-4,3)(-3,-4)(-3,4)(0,-5)(0,5)(3,-4)(3,4)(4,-3)(4,3)(5,0)     29
(-6,0)(0,-6)(0,6)(6,0)                                                     33
(-7,0)(0,-7)(0,7)(7,0)                                                     37
(-8,0)(0,-8)(0,8)(8,0)                                                     41
(-9,0)(0,-9)(0,9)(9,0)                                                     45
(-10,0)(-8,-6)(-8,6)(-6,-8)(-6,8)(0,-10)(0,10)(6,-8)(6,8)(8,-6)(8,6)(10,0) 57
(-11,0)(0,-11)(0,11)(11,0)                                                 61
(-12,0)(0,-12)(0,12)(12,0)                                                 65
		

Links

• Alexander Kritov, Source code

Crossrefs

Cf. A046080, A211432, A046109 (first differences), A349536 (in 1/8 sector).

Programs

C

/* See links */

PARI

f(n) = if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)); \\ A046109
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, Nov 27 2021

Formula

a(n) = (A211432(n) + 1)/2.

a(n) = a(n-1) + 4 + 8*A046080(n).