A280026 Fill an infinite square array by following a spiral around the origin; in the n-th cell, enter the number of earlier cells that can be seen from that cell.

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

Offset: 0

Author

N. J. A. Sloane, Dec 24 2016

Keywords

• nonn
• tabl

Comments

The spiral track being used here is the same as in A274640, except that the starting cell here is numbered 0 (as in A274641).

"Can be seen from" means "are on the same row, column, diagonal, or antidiagonal as".

The entry in a cell gives the number of earlier cells that are occupied in any of the eight cardinal directions. - Robert G. Wilson v, Dec 25 2016

First occurrence of k = 0,1,2,3,...: 0, 1, 2, 3, 5, 7, 8, 11, 14, 15, 19, 23, 24, 29, 34, 35, 41, 47, 48, 55, 62, ... - Robert G. Wilson v, Dec 25 2016

Examples

			The central portion of the spiral is:
.
    7---9---8---7---6
    |               |
    8   3---3---2   7
    |   |       |   |
    9   4   0---1   6
    |   |           |
   10   4---5---6---5
    |
    8---9--10--11--12 ...
		

Links

• Lars Blomberg, Table of n, a(n) for n = 0..10000

Crossrefs

See A280027 for an additive version.

Cf. A274640, A274641, A278354.

See A279211, A279212 for versions that follow antidiagonals in just one quadrant.

Programs

Mathematica

a[n_] := a[n - 1] + If[ IntegerQ@ Sqrt@ n || IntegerQ@ Sqrt[4n +1], 2 - Select[{Sqrt@ n, (Sqrt[4n +1] -1)/2}, IntegerQ][[1]], 1]; a[0] = 0; Array[a, 76, 0] (* Robert G. Wilson v, Dec 25 2016 *)

Formula

Empirically: a(0)=0, a(n+1)=a(n)+d for n>0, when n=k^2 or n=k*(k+1) then d=2-k, else d=1.

Extensions

Corrected a(23) and more terms from Lars Blomberg, Dec 25 2016