%I #23 Mar 20 2023 06:23:32

%S 1,1,1,1,2,2,3,2,2,3,2,2,3,2,2,3,1,3,3,1,4,1,4,3,5,5,1,4,3,4,5,4,4,5,

%T 6,6,7,4,4,5,5,6,2,4,1,4,5,1,6,2,6,4,6,5,5,7,2,3,4,6,5,5,7,2,3,8,1,4,

%U 3,6,7,5,5,3,5,7,6,3,1,1,7,8,7,7,4,5,8,5,9,6,6,8,7,7,6,8,9,9,3

%N Lexicographically earliest sequence of positive numbers on a square spiral such that no three equal numbers are collinear.

%C The first term a(1) = 1 lies at the (0,0) origin while all other terms lie on integer coordinates.

%H Scott R. Shannon, <a href="/A361486/b361486.txt">Table of n, a(n) for n = 1..10000</a>

%H Scott R. Shannon, <a href="/A361486/a361486_3.png">Image of the first 50000 terms on the square spiral</a>. The values are scaled across the spectrum from red to violet to show their relative size. Zoom in to see the numbers.

%H Scott R. Shannon, <a href="/A361486/a361486_4.png">Image highlighting the 1 valued terms in the first 50000 terms on the square spiral</a>. Zoom in to see the numbers.

%e a(5) = 2 as a(3) = 1 and a(4) = 1 lie on the horizontal line y = 1 relative to the starting square (assuming a counter-clockwise spiral) so a(5) cannot be 1.

%e a(7) = 3 as a(5) = 2 and a(6) = 2 lie on the vertical line x = -1 so a(7) cannot be 2, while a(1) = 1 and a(3) = 1 lie on the line y = x so a(7) cannot be 1.

%e a(21) = 4 as a(18) = 3 and a(19) = 3 lie on the line x = -2, a(6) = 2 and a(15) = 2 lie on the line y = 2*x + 2, while a(1) = 1 and a(3) = 1 lie on the line y = x, so a(21) cannot be 1, 2 or 3.

%Y Cf. A274640, A229037, A174344, A274923, A346294.

%K nonn,look

%O 1,5

%A _Scott R. Shannon_, Mar 13 2023