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A362249 Point number on a 4-arm square spiral of point n on the East arm scaled up by steps of that point itself.

%I A362249 #73 May 28 2023 08:46:07
%S A362249 1,4,19,16,13,64,149,58,81,70,139,324,583,268,217,256,233,244,569,
%T A362249 1024,1609,916,421,566,625,586,461,884,1591,2500,3611,2324,1323,1000,
%U A362249 1213,1296,1237,1048,1269,2284,3589,5184,7069,4924,3169,1804,1997,2290,2401,2318,2053,1724,3103,4876
%N A362249 Point number on a 4-arm square spiral of point n on the East arm scaled up by steps of that point itself.
%C A362249 Coordinates x=A340944(n), y=A340945(n) are the East arm of a 4-arm square spiral whose arms together visit each integer point in the plane. Call these arms the base spirals.
%C A362249 Construct a large spiral by taking point n on the East spiral as a vector u and scaling up the East spiral by that amount (so first step at u, then turn 90 degrees and step by distance |u|, and so on).
%C A362249 Point n along the large spiral falls somewhere on one of the base spirals. It is point number a(n) on base spiral number A362363(n).
%C A362249 In complex numbers, the East spiral is S(n) = A340944(n) + A340945(n)*i, the scale is u = S(n), the large spiral is L(t) = u*S(t), and its point n is at L(n) = S(n)^2 = S(k)*i^arm where a(n) = k and A362363(n) = arm.
%H A362249 Tamas Sandor Nagy, <a href="/A362249/a362249.pdf">Illustration for a(5)</a>.
%H A362249 Tamas Sandor Nagy, <a href="/A362249/a362249_1.pdf">Illustration for a(10)</a>.
%H A362249 Thomas Scheuerle, <a href="/A362249/a362249.png">Illustration of tracing the intersections in 2D</a>. It shows all four spirals with different colors. The green curve connects the intersection points where the rotated and scaled spiral intersects which one of the base spirals. The intersections at the cases of a(A002061(n)) are the local extreme values of the green curve. At a(n^2) our intersections are horizontally on the X axis.
%H A362249 Thomas Scheuerle, <a href="/A362249/a362249_1.png">The Cartesian distances of the intersections to the origin</a>. At a(n^2) this distance equals n^2, local minima. At a(A002061(n)) we reach local maxima of A001844(n-1).
%H A362249 Thomas Scheuerle, <a href="/A362249/a362249_2.png">Plot of this sequence in polar coordinates</a>. Angle = sqrt(n-1)*2*Pi, Radius = a(n).
%F A362249 A340944(a(n)) + i*A340945(a(n)) = (A340944(n) + i*A340945(n))^2 / i^A362363(n).
%F A362249 a(k^(2*n)) = k^(4*n).
%F A362249 a(4^n + 2^n) = 2^(4*n + 2).
%F A362249 a(A002061(n)) = 4*n^4 - 8*n^3 + 4*n^2 + 2*n - 1, for n > 0.
%e A362249 Explanatory diagrams for n = 5 and n = 10 are shown in the Links.
%o A362249 (PARI)
%o A362249 x(n, k) = (n^2 + k^2 - 2*n*k^2 + k^4)/(1 + k^2/(n - k^2)^2) - (k^2*(n^2 + k^2 - 2*n*k^2 + k^4))/((n - k^2)^2*(1 + k^2/(n - k^2)^2));
%o A362249 y(n, k) = (2*k*n^2)/((n - k^2)*(1 + k^2/(n - k^2)^2)) + (2*k^3)/((n - k^2)*(1 + k^2/(n - k^2)^2)) - (4*k^3*n)/(n - k^2 + (n*k^2)/(n^2 - 2*n*k^2 + k^4) - k^4/(n^2 - 2*n*k^2 + k^4)) + (2*k^5)/((n - k^2)*(1 + k^2/(n - k^2)^2));
%o A362249 t(n) =  {my(k = (sqrtint(4*n) + 1)\2); my(cy = abs(y(n,k))); my(cx = abs(x(n,k))); my(d = (cy > cx)); my(e = (n - k^2) < 0); return(max(cx,cy)^2+min(cx,cy)*(-1)^d*(-1)^e)};
%o A362249 a(n) = if(issquare(n), return(n^2), return(t(n)));
%o A362249 (MATLAB)
%o A362249 function a  = A362249( max_n )
%o A362249     E = [0 ; 0]; S = [0 ; 0]; W = [0 ; 0]; N = [0 ; 0]; V = [0 0];
%o A362249     for k = 1:4*max_n
%o A362249         l = V(1+mod(k+1,2)); s = (-1)^floor(k/2);
%o A362249         for m = l+(1*s):s:s*k
%o A362249             V(1+mod(k+1,2)) = m; V2 = V(end:-1:1).*[-1 1];
%o A362249             N = [N V2']; E = [E V']; S = [S -V2']; W = [W -V'];
%o A362249         end
%o A362249     end
%o A362249     for n = 2:max_n
%o A362249         [th,r] = cart2pol(E(1,n), E(2,n));
%o A362249         rot = [cos(-th) -sin(-th); sin(-th) cos(-th)];
%o A362249         v = E(:,n)'*rot*r;
%o A362249         jE = find(sum(abs([E(1,:)-v(1); E(2,:)-v(2)]),1) < 0.5);
%o A362249         jS = find(sum(abs([S(1,:)-v(1); S(2,:)-v(2)]),1) < 0.5);
%o A362249         jW = find(sum(abs([W(1,:)-v(1); W(2,:)-v(2)]),1) < 0.5);
%o A362249         jN = find(sum(abs([N(1,:)-v(1); N(2,:)-v(2)]),1) < 0.5);
%o A362249         a(n-1) = max([jE jS jW jN])-1;
%o A362249     end
%o A362249 end % _Thomas Scheuerle_, Apr 13 2023
%Y A362249 Cf. A000290, A001844, A002061, A340944, A340945, A362363.
%K A362249 nonn
%O A362249 1,2
%A A362249 _Tamas Sandor Nagy_ and _Thomas Scheuerle_, Apr 13 2023