A362249 - OEIS

%I #73 May 28 2023 08:46:07

%S 1,4,19,16,13,64,149,58,81,70,139,324,583,268,217,256,233,244,569,

%T 1024,1609,916,421,566,625,586,461,884,1591,2500,3611,2324,1323,1000,

%U 1213,1296,1237,1048,1269,2284,3589,5184,7069,4924,3169,1804,1997,2290,2401,2318,2053,1724,3103,4876

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

%C 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 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 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 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 Tamas Sandor Nagy, <a href="/A362249/a362249.pdf">Illustration for a(5)</a>.

%H Tamas Sandor Nagy, <a href="/A362249/a362249_1.pdf">Illustration for a(10)</a>.

%H 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 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 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 A340944(a(n)) + i*A340945(a(n)) = (A340944(n) + i*A340945(n))^2 / i^A362363(n).

%F a(k^(2*n)) = k^(4*n).

%F a(4^n + 2^n) = 2^(4*n + 2).

%F a(A002061(n)) = 4*n^4 - 8*n^3 + 4*n^2 + 2*n - 1, for n > 0.

%e Explanatory diagrams for n = 5 and n = 10 are shown in the Links.

%o (PARI)

%o 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 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 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 a(n) = if(issquare(n), return(n^2), return(t(n)));

%o (MATLAB)

%o function a  = A362249( max_n )

%o     E = [0 ; 0]; S = [0 ; 0]; W = [0 ; 0]; N = [0 ; 0]; V = [0 0];

%o     for k = 1:4*max_n

%o         l = V(1+mod(k+1,2)); s = (-1)^floor(k/2);

%o         for m = l+(1*s):s:s*k

%o             V(1+mod(k+1,2)) = m; V2 = V(end:-1:1).*[-1 1];

%o             N = [N V2']; E = [E V']; S = [S -V2']; W = [W -V'];

%o         end

%o     end

%o     for n = 2:max_n

%o         [th,r] = cart2pol(E(1,n), E(2,n));

%o         rot = [cos(-th) -sin(-th); sin(-th) cos(-th)];

%o         v = E(:,n)'*rot*r;

%o         jE = find(sum(abs([E(1,:)-v(1); E(2,:)-v(2)]),1) < 0.5);

%o         jS = find(sum(abs([S(1,:)-v(1); S(2,:)-v(2)]),1) < 0.5);

%o         jW = find(sum(abs([W(1,:)-v(1); W(2,:)-v(2)]),1) < 0.5);

%o         jN = find(sum(abs([N(1,:)-v(1); N(2,:)-v(2)]),1) < 0.5);

%o         a(n-1) = max([jE jS jW jN])-1;

%o     end

%o end % _Thomas Scheuerle_, Apr 13 2023

%Y Cf. A000290, A001844, A002061, A340944, A340945, A362363.

%K nonn

%O 1,2

%A _Tamas Sandor Nagy_ and _Thomas Scheuerle_, Apr 13 2023