A036705 -id:A036705 - cp's OEIS frontend

A232705 Number of Gaussian integers z satisfying (n-1)/2 < |z| < n/2.

0, 0, 4, 0, 8, 4, 8, 8, 20, 0, 16, 12, 24, 8, 28, 16, 28, 24, 40, 12, 32, 24, 44, 16, 48, 28, 48, 32, 52, 32, 40, 44, 64, 28, 72, 32, 76, 40, 72, 44, 56, 56, 84, 56, 80, 52, 88, 48, 92, 56, 92, 56, 96, 68, 88, 72, 108, 56, 104, 76, 112, 64, 124, 80, 104, 92, 112, 92, 120, 96, 116, 80, 144, 84

Offset: 1

Ralf Stephan, Nov 28 2013

nonn

Number of integer Cartesian grid points covered by a ring with width 1/2 and outer radius n/2.
For symmetry reasons, a(n) is a multiple of 4.
By bounds on the Gauss circle problem, a(n)/n -> Pi/2 as n -> infinity (see Wikipedia link). - Robert Israel and Peter Bala, Mar 26 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Plot of a(n)/(n*Pi/2) for n=1..10000
Wikipedia, Gauss circle problem
Index entries for Gaussian integers and primes

Cf. A047077, A036705, A333462.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for x from 1 to N/2 do
  for y from 1 to x do
    r:= 2*sqrt(x^2 + y^2);
    if r::integer then next fi;
    n:= ceil(r);
    if n > N then break fi;
    if x = y then V[n]:= V[n]+4 else V[n]:= V[n]+8 fi
od od:
convert(V,list); # Robert Israel, Mar 26 2020

a(n)=sum(i=-n,n,sum(j=-n,n,d=sqrt(i*i+j*j);if(d>(n-1)/2&&d

A036706 a(n)=number of Gaussian integers z=a+bi satisfying n - 1/2 < |z| <= n + 1/2, a>0, b>=0.

Original entry on oeis.org

0, 2, 3, 4, 8, 7, 10, 10, 12, 17, 14, 18, 17, 22, 22, 21, 28, 28, 28, 29, 28, 36, 35, 36, 36, 42, 41, 40, 46, 43, 50, 48, 47, 52, 56, 56, 57, 56, 62, 59, 66, 62, 66, 69, 66, 72, 69, 76, 76, 78, 79, 80, 86, 79, 84, 88, 85, 94, 84, 98, 95

Offset: 0

Clark Kimberling

nonn

Index entries for Gaussian integers and primes

a(n)=(1/4)*s(n) for n >= 1, where s=A036705.

cp's OEIS Frontend

A232705 Number of Gaussian integers z satisfying (n-1)/2 < |z| < n/2.

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