A275609 Square spiral in which each new term is the least nonnegative integer distinct from its (already assigned) eight neighbors.
0, 1, 2, 3, 1, 2, 1, 3, 2, 0, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 2, 0, 1, 3, 1, 2, 1, 2, 3, 0, 2, 3, 1, 3, 1, 2, 4, 1, 2, 1, 2, 1, 3, 1, 3, 2, 0, 2, 0, 3, 0, 3, 0, 1, 2, 1, 3, 1, 0, 2, 0, 4, 0, 1, 3, 0, 3, 0, 3, 0, 3, 0, 2, 0, 2, 0, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 3, 0, 3, 0, 2, 0, 2, 3, 1, 3, 1, 2, 3, 0, 2, 4, 1, 2
Offset: 0
Keywords
Examples
Illustration of initial terms as a spiral (n = 0..168): . . 1 - 2 - 1 - 0 - 4 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1 . | | . 3 0 - 3 - 2 - 1 - 3 - 1 - 3 - 2 - 0 - 2 - 0 2 . | | | | . 1 2 1 - 0 - 4 - 0 - 2 - 0 - 1 - 3 - 1 3 1 . | | | | | | . 0 4 3 2 - 1 - 3 - 1 - 3 - 2 - 0 2 0 2 . | | | | | | | | . 3 1 0 4 0 - 2 - 0 - 4 - 1 3 1 3 1 . | | | | | | | | | | . 0 2 3 1 3 1 - 3 - 2 0 2 0 2 0 . | | | | | | | | | | | | . 3 1 0 2 0 2 0 - 1 3 1 3 1 3 . | | | | | | | | | | | . 0 2 3 1 3 1 - 3 - 2 - 0 2 0 2 0 . | | | | | | | | | . 3 1 0 2 0 - 2 - 0 - 1 - 3 - 1 3 1 3 . | | | | | | | . 0 2 3 1 - 3 - 1 - 3 - 2 - 0 - 2 - 0 2 0 . | | | | | . 3 1 0 - 2 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1 3 . | | | . 0 2 - 3 - 1 - 3 - 1 - 3 - 2 - 0 - 2 - 0 - 2 - 0 . | . 1 - 4 - 0 - 2 - 0 - 2 - 0 - 1 - 3 - 1 - 3 - 1 - 3 . a(13) = 4 is the first "4" in the sequence and its four neighbors are 3 (southwest), 2 (south), 0 (southeast) and 1 (east) when a(13) is placed in the spiral. a(157) = 4 is the 6th "4" in the sequence and it is also the first "4" that is below the NE-SW main diagonal of the spiral (see the second term in the last row of the above diagram).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..100000 (first 5001 terms from N. J. A. Sloane)
- F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
- Rémy Sigrist, Colored illustration of the sequence (with cells (x,y) such that -100 <= x <= 100 and -100 <= y <= 100)
- N. J. A. Sloane, Central portion of spiral, shown without spaces [The central 0 (at (0,0)) has been changed to an X. The illustration shows cells (x,y) with -35 <= x <= 35, -33 <= y <= 36.]
Crossrefs
Programs
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Maple
fx:= proc(n) option remember; `if`(n=1, 0, (k-> fx(n-1)+sin(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4)) end: fy:= proc(n) option remember; `if`(n=1, 0, (k-> fy(n-1)-cos(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4)) end: b:= proc() -1 end: a:= proc(n) option remember; local x, y, s, m; x, y:= fx(n+1), fy(n+1); if n>0 then a(n-1) fi; if b(x, y) >= 0 then b(x, y) else s:= {b(x+1, y+1), b(x-1, y-1), b(x+1, y-1), b(x-1, y+1), b(x+1, y ), b(x-1, y ), b(x , y+1), b(x , y-1)}; for m from 0 while m in s do od; b(x, y):= m fi end: seq(a(n), n=0..120); # Alois P. Heinz, Mar 29 2019 -
Mathematica
fx[n_] := fx[n] = If[n == 1, 0, Function[k, fx[n - 1] + Sin[k*Pi/2]][Mod[ Floor[Sqrt[4*(n - 2) + 1]], 4]]]; fy[n_] := fy[n] = If[n == 1, 0, Function[k, fy[n - 1] - Cos[k*Pi/2]][Mod[ Floor[Sqrt[4*(n - 2) + 1]], 4]]]; b[, ] := -1; a[n_] := a[n] = Module[{x, y, s, m}, {x, y} = {fx[n + 1], fy[n + 1]}; If[n > 0, a[n - 1]]; If [b[x, y] >= 0, b[x, y], s = {b[x + 1, y + 1], b[x - 1, y - 1], b[x + 1, y - 1], b[x - 1, y + 1], b[x + 1, y], b[x - 1, y], b[x, y + 1], b[x, y - 1]}; For[m = 0, MemberQ[s, m], m++]; b[x, y] = m]]; a /@ Range[0, 120] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)
Formula
a(n) = A274917(n) - 1.
Comments