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

%I A232705 #26 Mar 27 2020 06:39:24
%S A232705 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,
%T A232705 48,32,52,32,40,44,64,28,72,32,76,40,72,44,56,56,84,56,80,52,88,48,92,
%U A232705 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
%N A232705 Number of Gaussian integers z satisfying (n-1)/2 < |z| < n/2.
%C A232705 Number of integer Cartesian grid points covered by a ring with width 1/2 and outer radius n/2.
%C A232705 For symmetry reasons, a(n) is a multiple of 4.
%C A232705 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
%H A232705 Robert Israel, <a href="/A232705/b232705.txt">Table of n, a(n) for n = 1..10000</a>
%H A232705 Robert Israel, <a href="/A232705/a232705.png">Plot of a(n)/(n*Pi/2) for n=1..10000</a>
%H A232705 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gauss_circle_problem">Gauss circle problem</a>
%H A232705 <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%p A232705 N:= 100: # for a(1)..a(N)
%p A232705 V:= Vector(N):
%p A232705 for x from 1 to N/2 do
%p A232705   for y from 1 to x do
%p A232705     r:= 2*sqrt(x^2 + y^2);
%p A232705     if r::integer then next fi;
%p A232705     n:= ceil(r);
%p A232705     if n > N then break fi;
%p A232705     if x = y then V[n]:= V[n]+4 else V[n]:= V[n]+8 fi
%p A232705 od od:
%p A232705 convert(V,list); # _Robert Israel_, Mar 26 2020
%o A232705 (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)))
%Y A232705 Cf. A047077, A036705, A333462.
%K A232705 nonn
%O A232705 1,3
%A A232705 _Ralf Stephan_, Nov 28 2013