A331363 Pairs of coordinates of the corners in a counterclockwise triangular spiral.

0, 0, 1, 0, 0, 1, -2, -1, 3, -1, 0, 2, -4, -2, 5, -2, 0, 3, -6, -3, 7, -3, 0, 4, -8, -4, 9, -4, 0, 5, -10, -5, 11, -5, 0, 6, -12, -6, 13, -6, 0, 7, -14, -7, 15, -7, 0, 8, -16, -8, 17, -8, 0, 9, -18, -9, 19, -9, 0, 10, -20, -10, 21, -10, 0, 11

Offset

1,7

Comments

Odd n yields the x- and even n the y-coordinates (i.e., x- and y-coordinates alternate in the sequence).

Links

Table of n, a(n) for n=1..66.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Formula

a(n) = ceiling(1/18*n*((2 - n) mod 6 + 4*n mod (-3) + 1)), for n >= 1.

x(n) = ceiling(n - 2/3*(n^2 + 1) mod 3), for n >= 1 (x-coordinates).

y(n) = floor(2*n/3)*((2 - n) mod (-3) + 1), for n >= 1 (y-coordinates).

From Colin Barker, May 03 2020: (Start)

G.f.: x^3*(1 + x^3 - 2*x^4 - x^5 + x^6 - x^7) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).

a(n) = 2*a(n-6) - a(n-12) for n>12.

(End)

From Wolfdieter Lang, Jul 13 2020: (Start)

Bisection: x(k) = a(2*k-1). x(3+3*l) = 0, x(1+3*l) = -2*l, x(2+3*l) = 1+2*l, for l >= 0.

x(k) = (2*(k-1)*modp((k-4)^2,3) - (2*k-1)*modp((k-2)^2,3) + 1)/3, for k >= 1.

y(k) = a(2*k). y(3+3*k) = 1+l, y(1+3*k) = -l = y(2+3*k), for l >= 0.

y(k) = ((k-1)*modp((k-1)^2,3) + (k-2)*modp((k+1)^2,3) - k*modp(k^2,3) -(k-3))/3, k >= 1.

G.f.s: Gx(t) = t^2*(1 - 2*t^2 + t^3)/(1 - t^3)^2, and Gy(t) = t^3*(1 - t - t^2) / (1 - t^3)^2.

This produces the g.f. G(x) = Gy(x^2) + Gx(x^2)/x given by Colin Barker.

(End)

Example

X- and y-coordinates of the corners alternate in the sequence: 0, 0, 1, 0, 0, 1, -2,-1, 3, -1, ...
                      (0,4)
                     .     \
                    .       \
                   .         \
                      (0,3)   \
                     /     \   \
                    /       \   \
                   /         \   \
                  /   (0,2)   \   \
                 /   /     \   \   \
                /   /       \   \   \
               /   /         \   \   \
              /   /   (0,1)   \   \   \
             /   /   /     \   \   \   \
            /   /   /       \   \   \   \
           /   /   /         \   \   \   \
          /   /   /   (0,0)->(1,0)\   \   \
         /   /   /                 \   \   \
        /   /   /                   \   \   \
       /   /  (-2,-1)------------->(3,-1)\   \
      /   /                               \   \
     /   /                                 \   \
    /  (-4,-2)--------------------------->(5,-2)\
   /                                             \
  /                                               \
(-6,-3)------------------------------------------>(7,-3)

Mathematica

a[n_] := Ceiling[1/18*n*(Mod[2 - n, 6] + 4*Mod[n, -3] + 1)]; Table[ a[n], {n, 66}] (* or *)
LinearRecurrence[{0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1}, {0, 0, 1, 0, 0, 1, -2, -1, 3, -1, 0, 2}, 66]

Crossrefs

Cf. A329116, A329972.

Sequence in context: A186976 A160550 A172038 * A359368 A353490 A226131

Adjacent sequences: A331360 A331361 A331362 * A331364 A331365 A331366

Keyword

sign,look

Author

Mikk Heidemaa, May 03 2020

Status

approved