A362249 -id:A362249 - cp's OEIS frontend

A362363 Arm number of the base spiral in A362249 which visits large spiral point n there.

0, 0, 2, 3, 0, 0, 0, 1, 0, 1, 2, 0, 2, 2, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 3, 0, 3, 0, 3, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 2, 2, 2, 2, 0, 3, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 2

Offset: 1

Tamas Sandor Nagy and Thomas Scheuerle, Apr 17 2023

nonn

Arms are numbered 0,1,2,3 for the base spirals with first segment directed East, South, West, North, respectively.

This numbering is successive arms around in the same direction that the spirals themselves turn (both clockwise in the diagrams in A362249).

			a(5) = 0 because A362249(5) = 13 that is on spiral "E", which is encoded here as 0.
a(8) = 1 because A362249(8) = 58 that is on spiral "S", which is encoded here as 1.
a(11) = 2 because A362249(11) = 139 that is on spiral "W", which is encoded here as 2.
a(34) = 3 because A362249(34) = 1000 that is on spiral "N", which is encoded here as 3.

Cf. A362249, A362265 (indices of 0's).

function a  = A362363( 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) = find([length(jE) length(jS) length(jW) length(jN)] > 0) - 1;
    end
end % Thomas Scheuerle, Apr 19 2023

If n is a square:

a(n) = 3*(n+1 mod 2); (a(n) = 3 for even squares).

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A362363 Arm number of the base spiral in A362249 which visits large spiral point n there.

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