A397719 Shortest path indices during walk over parabolic (exponent = 2) spiral lattice with divergence angle equal to golden angle and starting at the center.

0, 1, 2, 3, 4, 6, 5, 8, 11, 9, 7, 10, 13, 18, 15, 12, 17, 14, 19, 16, 21, 26, 31, 23, 20, 25, 30, 22, 27, 32, 24, 29, 34, 42, 37, 45, 40, 35, 43, 38, 33, 28, 36, 44, 39, 47, 55, 63, 50, 58, 66, 53, 61, 48, 56, 64, 51, 59, 46, 41, 49, 57, 65, 52, 60, 68, 76, 84, 71, 79, 92, 105

Offset

0,3

Comments

Start with spiral lattice generated by (k^2 * cos(k*theta), k^2 * sin(k*theta)) for theta being the golden angle (A131988) and k = 0, 1, 2, 3, ... .

Next, starting at point (0,0), move to the closest unvisited point. a(n) is the index k corresponding to the point reached after n moves.

Will all points of the lattice be visited?

Links

Table of n, a(n) for n=0..71.

Example

Below is table of Cartesian coordinates of the first 10 points:
  k   x(k)   y(k)
  ---------------
  0   0.00   0.00
  1  -0.74   0.68
  2   0.35  -3.98
  3   5.48   7.14
  4 -15.76  -2.79
  5  21.09 -13.42
  6  -9.35  34.77
  7 -22.58 -43.48
  8  60.12  21.95
  9 -74.87  30.91
We start at point (0.00, 0.00) that corresponds to k = 0, therefore a(0) = 0.
Unvisited point closest to (0.00, 0.00) is (-0.74, 0.68) corresponding to k = 1, therefore a(1) = 1.
Next points in the sequence correspond to k = 2, 3, and 4.
Unvisited point closest to k = 4 (-15.76, -2.79) is (-9.35, 34.77) corresponding to k = 6, therefore a(5) = 6.
Unvisited point closest to k = 6 (-9.35, 34.77) is (21.09, -13.43) corresponding to k = 5, therefore a(6) = 5.

Crossrefs

Cf. A000796, A001622, A131988, A396911, A397408.

Sequence in context: A032447 A224531 A058213 * A348711 A080997 A151942

Adjacent sequences: A397716 A397717 A397718 * A397720 A397723 A397726

Keyword

nonn

Author

Witold Tatkiewicz, Jul 06 2026

Status

approved