This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137243 #36 Apr 09 2023 02:31:29
%S A137243 8,16,32,48,80,96,144,176,224,256,336,368,464,512,576,640,768,816,960,
%T A137243 1024,1120,1200,1376,1440,1600,1696,1840,1936,2160,2224,2464,2592,
%U A137243 2752,2880,3072,3168,3456,3600,3792,3920,4240,4336,4672,4832,5024,5200,5568
%N A137243 Number of coprime pairs (a,b) with -n <= a,b <= n.
%C A137243 Number of square lattice points in the region {(x, y): |x| <= n, |y| <= n} visible from the origin. - _Paolo Xausa_, Mar 25 2023
%H A137243 Paolo Xausa, <a href="/A137243/b137243.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Seiichi Manyama).
%H A137243 Tom M. Apostol, <a href="https://doi.org/10.1007/978-1-4757-5579-4">Introduction to Analytic Number Theory</a>, Springer, New York, NY, 1976, pp. 62-64.
%F A137243 a(n) = 4*A018805(n) + 4. - _Charles R Greathouse IV_, Aug 06 2012
%F A137243 a(n) = a(n-1) + 8*EulerPhi(n), with a(1)=8. - _Juan M. Marquez_, Apr 24 2015
%F A137243 a(n) ~ (24/Pi^2)*n^2 = 4*A059956*n^2. - _Paolo Xausa_, Mar 25 2023
%e A137243 a(1) = 8 because there are eight coprime pairs (1,1), (1,0), (1,-1), (0,1), (0,-1), (-1,1), (-1,0), (-1,-1) with integral entries of absolute value <= 1.
%e A137243 In the same way a(2) = 16 as there are sixteen coprime pairs (2,1), (2,-1), (1,2), (1,1), (1,0), (1,-1), (1,-2), (0,1), (0,-1), (-1,2), (-1,1), (-1,0), (-1,-1), (-1,-2), (-2,1), (-2,-1) of integral entries of absolute value <= 2.
%t A137243 A137243[nmax_]:=8Accumulate[EulerPhi[Range[nmax]]];A137243[100] (* _Paolo Xausa_, Mar 25 2023 *)
%o A137243 (PARI) a(n)=4*sum(k=1, n, moebius(k)*(n\k)^2)+4 \\ _Charles R Greathouse IV_, Aug 06 2012
%o A137243 (Python)
%o A137243 from functools import lru_cache
%o A137243 @lru_cache(maxsize=None)
%o A137243 def A137243(n):
%o A137243 if n == 0:
%o A137243 return 0
%o A137243 c, j = 0, 2
%o A137243 k1 = n//j
%o A137243 while k1 > 1:
%o A137243 j2 = n//k1 + 1
%o A137243 c += (j2-j)*(A137243(k1)//4-1)
%o A137243 j, k1 = j2, n//j2
%o A137243 return 4*(n*(n-1)-c+j) # _Chai Wah Wu_, Mar 29 2021
%Y A137243 Cf. A018805, A059956.
%K A137243 nonn
%O A137243 1,1
%A A137243 Kilian Kilger (kilian(AT)mathi.uni-heidelberg.de), May 11 2008