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

1,3

Comments

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

Links

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

Maple

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

Prog

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

Crossrefs

Cf. A047077, A036705, A333462.

Sequence in context: A244336 A137323 A021075 * A245174 A109169 A011291

Adjacent sequences: A232702 A232703 A232704 * A232706 A232707 A232708

Keyword

nonn

Author

Ralf Stephan, Nov 28 2013

Status

approved