A280026 - OEIS

%I #36 Dec 28 2016 13:50:29

%S 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,

%T 10,11,12,13,14,15,11,12,13,14,15,16,12,13,14,15,16,17,18,13,14,15,16,

%U 17,18,19,14,15,16,17,18,19,20,21,15,16,17,18,19,20,21

%N 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.

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

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

%C 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

%C 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

%H Lars Blomberg, <a href="/A280026/b280026.txt">Table of n, a(n) for n = 0..10000</a>

%F 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.

%e The central portion of the spiral is:

%e .

%e     7---9---8---7---6

%e     |               |

%e     8   3---3---2   7

%e     |   |       |   |

%e     9   4   0---1   6

%e     |   |           |

%e    10   4---5---6---5

%e     |

%e     8---9--10--11--12 ...

%t 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 *)

%Y See A280027 for an additive version.

%Y Cf. A274640, A274641, A278354.

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

%K nonn,tabl

%O 0,3

%A _N. J. A. Sloane_, Dec 24 2016

%E Corrected a(23) and more terms from _Lars Blomberg_, Dec 25 2016