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A274640
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Counterclockwise square spiral constructed by greedy algorithm, so that each row, column, and diagonal contains distinct numbers.
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52
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1, 2, 3, 4, 2, 3, 4, 5, 6, 1, 4, 6, 2, 1, 6, 5, 3, 1, 5, 2, 6, 1, 2, 4, 5, 3, 7, 8, 5, 4, 9, 7, 8, 3, 10, 11, 4, 7, 8, 6, 3, 9, 5, 7, 8, 9, 10, 11, 12, 6, 8, 9, 11, 10, 12, 13, 7, 6, 10, 9, 12, 13, 14, 15, 8, 2, 9, 12, 7, 10, 11, 13, 14, 10, 9, 6, 13, 5, 3, 15, 16, 7, 1, 10, 13, 12, 14, 11, 15, 3, 8, 5, 1, 12, 11, 14, 7, 4, 2, 16, 9, 17, 1, 8, 11
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OFFSET
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0,2
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COMMENTS
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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
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).
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LINKS
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EXAMPLE
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The spiral begins:
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9--16---2---4---7--14--11--12---1---5---8
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17 8--15--14--13--12---9--10---6---7 3
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1 2 4--11--10---3---8---7---9 13 15
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8 9 7 3---5---6---1---2 4 12 11
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11 12 8 1 2---4---3 6 5 10 14
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15 7 6 5 3 1---2 4 8 11 12
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14 10 3 2 4---5---6---1 7 9 13
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7 11 9 6---1---2---4---5---3 8 10
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4 13 5---7---8---9--10--11--12---6 1
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12 14--10---9---6--13---5---3--15--16---7
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10--15---1--12--16---8--14--13--11--18--17
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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[]]];
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CROSSREFS
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In the same spirit as the infinite Sudoku array A269526.
Cf. A274821 (the same construction on a hexagonal tiling).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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