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

0,2

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.

Links

Table of n, a(n) for n=0..60.

Alexander Kritov, Source code

Formula

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

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

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A314700 A032493 A362256 * A334524 A125018 A074787

Adjacent sequences: A349535 A349536 A349537 * A349539 A349540 A349541

Keyword

nonn

Author

Alexander Kritov, Nov 21 2021

Status

approved