OFFSET
0,2
COMMENTS
Presumably every row, column, and diagonal is a permutation of the natural numbers, but is there a proof? - N. J. A. Sloane, Jul 10 2016
The n-th cell in the spiral has coordinates x = A174344(n+1), y = A274923(n+1). - N. J. A. Sloane, Jul 11 2016
From Robert G. Wilson v, Dec 25 2016: (Start) [Memo: all these numbers need to decreased by 1, since the offset here is 0. See A324481. - N. J. A. Sloane, Jul 23 2017. Furthermore, the numbers don't seem correct, even after subtracting 1. - N. J. A. Sloane, Jul 04 2019]
Index of first appearance of k = 1,2,3,...: 1, 2, 3, 7, 8, 15, 17, 25, 35, 41, 47, 61, 62, 89, 98, 99, 121, 129, 130, 143, 197, 208, 225, 239, 271, ..., .
1 appears at: 1, 4, 12, 19, 22, 33, 42, 68, 79, 120, 179, 194, 302, 311, 445, 489, 511, 558, 630, 708, 847, 877, 907, ..., .
2 appears at: 2, 5, 9, 16, 48, 52, 70, 73, 88, 95, 110, 146, 280, 291, 309, 327, 488, 605, 656, 681, 735, 778, 1000, ..., .
3 appears at: 3, 6, 10, 23, 29, 36, 56, 76, 97, 105, 153, 168, 184, 252, 338, 437, 457, 670, 818, 906, 953, 967, ..., . (End).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..20000
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.
Alois P. Heinz, Distribution of a(n) for n <= 4010000
Kerry Mitchell, Color-coded version of spiral, (1): the colors represent the values, from black (small) to white (large) (jpg file, low resolution)
Kerry Mitchell, Color-coded version of spiral, (1a): the colors represent the values, from black (small) to white (large) (tiff file, much higher resolution)
Kerry Mitchell, Color-coded version of spiral, (2): values <= 100 are black and those > 100 are white.
Zak Seidov, Distribution of a(n) for first 20001 terms
EXAMPLE
The spiral begins:
.
9--16---2---4---7--14--11--12---1---5---8
| |
17 8--15--14--13--12---9--10---6---7 3
| | | |
1 2 4--11--10---3---8---7---9 13 15
| | | | | |
8 9 7 3---5---6---1---2 4 12 11
| | | | | | | |
11 12 8 1 2---4---3 6 5 10 14
| | | | | | | | | |
15 7 6 5 3 1---2 4 8 11 12
| | | | | | | | |
14 10 3 2 4---5---6---1 7 9 13
| | | | | | |
7 11 9 6---1---2---4---5---3 8 10
| | | | |
4 13 5---7---8---9--10--11--12---6 1
| | |
12 14--10---9---6--13---5---3--15--16---7
|
10--15---1--12--16---8--14--13--11--18--17
.
E: 1, 2, 4, 8, 11, 12, 16, 9, 19, 24, 22, ...
NE: 1, 3, 2, 9, 7, 8, 12, 15, 13, 17, 20, ...
N: 1, 4, 6, 3, 12, 14, 15, 18, 20, 26, 25, ...
NW: 1, 2, 3, 4, 8, 9, 7, 11, 14, 10, 22, ...
W: 1, 3, 5, 6, 7, 15, 10, 17, 13, 25, 14, ...
SW: 1, 4, 6, 5, 14, 10, 11, 23, 16, 18, 21, ...
S: 1, 5, 2, 9, 13, 8, 7, 11, 10, 17, 19, ...
SE: 1, 6, 5, 12, 16, 17, 21, 24, 27, 13, 15, ...
MAPLE
# Maple program from Alois P. Heinz, Jul 12 2016:
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() 0 end:
a:= proc(n) local x, y, s, i, t, m;
x, y:= fx(n+1), fy(n+1);
if b(x, y) > 0 then b(x, y)
else s:={};
for i do t:=b(x+i, y+i); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x-i, y-i); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x+i, y-i); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x-i, y+i); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x+i, y ); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x-i, y ); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x , y+i); if t>0 then s:=s union {t} else break fi od;
for i do t:=b(x , y-i); if t>0 then s:=s union {t} else break fi od;
for m while m in s do od;
b(x, y):= m
fi
end:
seq(a(n), n=0..1000);
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]]]; Clear[b]; b[_, _] = 0; a[n_] := Module[{x, y, s, i, t, m}, {x, y} = {fx[n+1], fy[n+1]}; If[b[x, y] > 0, b[x, y], s = {};
For[i=1, True, i++, t=b[x+i, y+i]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x-i, y-i]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x+i, y-i]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x-i, y+i]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x+i, y ]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x-i, y ]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x , y+i]; If[t>0, s=Union[s, {t}], Break[]]];
For[i=1, True, i++, t=b[x , y-i]; If[t>0, s=Union[s, {t}], Break[]]];
m = 1; While[MemberQ[s, m], m++]; b[x, y] = m]]; Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Nov 14 2016, after Alois P. Heinz *)
PROG
(Python)
class Lines: # manage lines in direction d = dx + dy*1j
def __init__(self, d):
self.lines={}; self.t = d.real/d.imag if d.imag else None
def __call__(self, pos): # Return the line through pos in direction d
index = pos.imag if self.t is None else pos.real - pos.imag*self.t
if index not in self.lines: self.lines[index] = Values()
return self.lines[index]
class Values(set): # the set of used numbers on a given line
def next(self, n): # return least k >= n not on this line
return min(m+1 for m in self if m+1 >= n and m+1 not in self
) if n in self else n
def A274640(): # generator of the sequence, see below for possible usage
lines = [Lines(d) for d in (1, 1+1j, 1j, 1-1j)]; pos = 0
for side in range(9**9):
for _ in range(side//2 + 1):
n = 1; lines_here = [L(pos) for L in lines]
while any(n < (n := L.next(n)) for L in lines_here): pass
yield n; any(L.add(n) for L in lines_here); pos += 1j**side
[a for a, _ in zip(A274640(), range(99))] # M. F. Hasler, Feb 01 2025
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Zak Seidov and Kerry Mitchell, Jun 30 2016
EXTENSIONS
Corrected and extended by Alois P. Heinz, Jul 12 2016
STATUS
approved