A362363 -id:A362363 - cp's OEIS frontend

A362265 Indices m for which A362363(m) = 0, meaning the large spiral point in A362249 falls on the East base spiral.

1, 2, 5, 6, 7, 9, 12, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 30, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 51, 52, 54, 56, 61, 63, 65, 67, 68, 69, 70, 71, 72, 73, 75, 77, 79, 81, 83, 86, 88, 90, 97, 99, 101, 103, 105, 106, 107, 108, 109, 110, 111, 113, 115, 117, 119, 121, 123, 125, 126, 128, 130

Offset: 1

Tamas Sandor Nagy and Thomas Scheuerle, Apr 13 2023

nonn

If m is a term then further terms can be found by writing m = s^2 + r such that s^2 is the square closest to m (and r is positive or negative). Then further terms are k = (t*s)^2 + t*r for odd t (but only sometimes even t).

			6 is a term since in A362249, its n=6 large spiral point 6 falls on its East base spiral.

Thomas Scheuerle, Spiral of dots. Each dot corresponds to a spiral from A362249. If this spiral meets with spiral "E", the color is blue. Other colors: "S" = orange, "W" = yellow, "N" = violet. Spirals where n in A362249 is an even square number are located on the x axis extending from the midpoint to the right. The odd square numbers extend to the left.

Cf. A362249, A000290, A001844, A002061.

All numbers of the form (2*k+1)^2 will be found inside this sequence but not (2*k)^2.

All numbers of the form 4^k+2^k, 4*k^2+k and k > 0, 9*(2*k+1)^2-4*k-2, 9*k^2+3*k and k > 0, 16*(2*k+1)^2+2*k+1 will be found inside this sequence.

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

Original entry on oeis.org

1, 4, 19, 16, 13, 64, 149, 58, 81, 70, 139, 324, 583, 268, 217, 256, 233, 244, 569, 1024, 1609, 916, 421, 566, 625, 586, 461, 884, 1591, 2500, 3611, 2324, 1323, 1000, 1213, 1296, 1237, 1048, 1269, 2284, 3589, 5184, 7069, 4924, 3169, 1804, 1997, 2290, 2401, 2318, 2053, 1724, 3103, 4876

Offset: 1

Tamas Sandor Nagy and Thomas Scheuerle, Apr 13 2023

nonn

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.
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).
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).
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.

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

Tamas Sandor Nagy, Illustration for a(5).
Tamas Sandor Nagy, Illustration for a(10).
Thomas Scheuerle, Illustration of tracing the intersections in 2D. 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.
Thomas Scheuerle, The Cartesian distances of the intersections to the origin. At a(n^2) this distance equals n^2, local minima. At a(A002061(n)) we reach local maxima of A001844(n-1).
Thomas Scheuerle, Plot of this sequence in polar coordinates. Angle = sqrt(n-1)*2*Pi, Radius = a(n).

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

function a  = A362249( max_n )
    E = [0 ; 0]; S = [0 ; 0]; W = [0 ; 0]; N = [0 ; 0]; V = [0 0];
    for k = 1:4*max_n
        l = V(1+mod(k+1,2)); s = (-1)^floor(k/2);
        for m = l+(1*s):s:s*k
            V(1+mod(k+1,2)) = m; V2 = V(end:-1:1).*[-1 1];
            N = [N V2']; E = [E V']; S = [S -V2']; W = [W -V'];
        end
    end
    for n = 2:max_n
        [th,r] = cart2pol(E(1,n), E(2,n));
        rot = [cos(-th) -sin(-th); sin(-th) cos(-th)];
        v = E(:,n)'*rot*r;
        jE = find(sum(abs([E(1,:)-v(1); E(2,:)-v(2)]),1) < 0.5);
        jS = find(sum(abs([S(1,:)-v(1); S(2,:)-v(2)]),1) < 0.5);
        jW = find(sum(abs([W(1,:)-v(1); W(2,:)-v(2)]),1) < 0.5);
        jN = find(sum(abs([N(1,:)-v(1); N(2,:)-v(2)]),1) < 0.5);
        a(n-1) = max([jE jS jW jN])-1;
    end
end % Thomas Scheuerle, Apr 13 2023

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

A340944(a(n)) + i*A340945(a(n)) = (A340944(n) + i*A340945(n))^2 / i^A362363(n).
a(k^(2*n)) = k^(4*n).
a(4^n + 2^n) = 2^(4*n + 2).
a(A002061(n)) = 4*n^4 - 8*n^3 + 4*n^2 + 2*n - 1, for n > 0.

cp's OEIS Frontend

A362265 Indices m for which A362363(m) = 0, meaning the large spiral point in A362249 falls on the East base spiral.

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