A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy
• nice
• changed

Comments

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012

The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

Examples

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Michael Gilleland, Some Self-Similar Integer Sequences
• Eric Weisstein's World of Mathematics, Circle Lattice Points

Crossrefs

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.

Cf. A000328, A051132, A009177.

Programs

Haskell

a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012

Maple

N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015

Mathematica

Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)

PARI

a(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)) \\ Charles R Greathouse IV, Feb 01 2017

PARI

a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017

Python

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

Formula

a(n) = A000328(n) - A051132(n).

a(n) = 8*A046080(n) + 4 for n > 0.

a(n) = A004018(n^2).

From Jean-Christophe Hervé, Dec 01 2013: (Start)

a(A084647(k)) = 28.

a(A084648(k)) = 36.

a(A084649(k)) = 44. (End)

a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017

a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

From Hugo Pfoertner, Sep 21 2023: (Start)

a(n) = 8*A063014(n) - 4 for n > 0.

a(n) = 4*A256452(n) for n > 0. (End)