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

0,2

Comments

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

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

Formula

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

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

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

a(A084647(k)) = 28. - Jean-Christophe Hervé, Dec 01 2013

a(A084648(k)) = 36. - Jean-Christophe Hervé, Dec 01 2013

a(A084649(k)) = 44. - Jean-Christophe Hervé, Dec 01 2013

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 for n > 0.(End)

Example

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.

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 *)

Prog

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

Crossrefs

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

Cf. A000328, A051132.

Sequence in context: A295643 A190718 A035621 * A294246 A107680 A358509

Adjacent sequences: A046106 A046107 A046108 * A046110 A046111 A046112

Keyword

nonn,easy,nice,changed

Author

Eric W. Weisstein

Status

approved