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 A345427 #18 Mar 29 2023 09:19:06
%S A345427 1,3,5,8,10,14,15,20,21,24,26,33,27,34,35,38,41,41,33,45,37,41,46,63,
%T A345427 36,31,31,25,35,50,39,56,23,15,25,14,-6,8,-5,-3,-6,3,-49,6,-6,-15,-8,
%U A345427 -9,-78,-124,-112,-100,-118,-122,-133,-109,-110,-139,-127,-117,-237,-166,-185,-218,-171,-215
%N A345427 For 1<=x<=n, 1<=y<=n, write gcd(x,y) = u*x+v*y with u,v minimal; a(n) = sum of the values of v.
%C A345427 Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when x=y.
%t A345427 T[x_, y_] := T[x, y] = Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= x^2 + y^2 && u*x + v*y == GCD[x, y], {u, v}, Integers], #.# &]];
%t A345427 a[n_] := a[n] = Sum[T[x, y][[1, 2]], {x, 1, n}, {y, 1, n}];
%t A345427 Table[Print[n, " ", a[n]]; a[n], {n, 1, 62}] (* _Jean-François Alcover_, Mar 28 2023 *)
%o A345427 (Python)
%o A345427 from sympy.core.numbers import igcdex
%o A345427 def A345427(n): return sum(v for u, v, w in (igcdex(x,y) for x in range(1,n+1) for y in range(1,n+1))) # _Chai Wah Wu_, Jun 22 2021
%Y A345427 Cf. A345415-A345428.
%K A345427 sign
%O A345427 1,2
%A A345427 _N. J. A. Sloane_, Jun 22 2021