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

0,3

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

Links

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

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.

Example

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 ...

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 *)

Crossrefs

See A280027 for an additive version.

Cf. A274640, A274641, A278354.

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

Sequence in context: A218535 A306592 A319246 * A323080 A194806 A229790

Adjacent sequences: A280023 A280024 A280025 * A280027 A280028 A280029

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 24 2016

Extensions

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

Status

approved