OFFSET
1,2
COMMENTS
If we only worry about queens' moves then we get the array in A269526.
Presumably, as in A269526, every column, every row, and every diagonal is a permutation of the natural numbers.
The knights only affect the squares in their immediate neighborhood, so this array will have very similar properties to A269526. The most noticeable difference is that the first column is no longer A000027, it is now A274631.
A piece that can move like a queen or a knight is known as a Maharaja. If we subtract 1 from the entries here we obtain A308201. - N. J. A. Sloane, Jun 30 2019
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..10010
EXAMPLE
The array begins:
1, 3, 6, 2, 7, 5, 8, 4, 9, 10, 15, 13, 11, 18, 12, 20, 16, 22, ...
2, 5, 8, 4, 1, 9, 6, 11, 3, 12, 7, 14, 17, 15, 10, 13, 19, 24, ...
4, 7, 9, 11, 3, 10, 13, 14, 1, 2, 8, 5, 6, 16, 22, 17, 21, 12, ...
3, 1, 10, 6, 2, 7, 12, 5, 15, 4, 16, 20, 13, 9, 11, 14, 25, 8, ...
5, 2, 12, 13, 4, 1, 9, 3, 6, 11, 10, 17, 19, 8, 7, 15, 23, 29, ...
6, 4, 11, 3, 8, 14, 10, 16, 13, 1, 2, 7, 15, 5, 24, 21, 9, 28, ...
7, 9, 1, 5, 6, 11, 2, 12, 8, 14, 3, 21, 23, 22, 4, 27, 18, 30, ...
8, 12, 2, 7, 9, 15, 1, 19, 4, 5, 6, 10, 18, 3, 26, 23, 11, 31, ...
10, 6, 3, 14, 12, 4, 5, 9, 11, 7, 1, 8, 16, 13, 2, 24, 28, 20, ...
9, 13, 4, 1, 10, 2, 7, 18, 12, 3, 17, 19, 24, 14, 20, 5, 8, 6, ...
11, 8, 5, 9, 13, 3, 15, 1, 2, 6, 20, 18, 10, 4, 17, 7, 12, 14, ...
12, 10, 7, 18, 11, 6, 4, 8, 14, 9, 5, 15, 21, 2, 16, 26, 3, 13, ...
13, 15, 17, 12, 14, 16, 18, 7, 10, 22, 11, 3, 8, 19, 23, 9, 2, 1, ...
14, 11, 19, 8, 5, 20, 3, 2, 16, 13, 12, 25, 4, 10, 6, 18, 7, 15, ...
16, 18, 21, 10, 15, 13, 11, 17, 5, 8, 9, 6, 7, 30, 25, 28, 20, 19, ...
15, 20, 13, 17, 16, 12, 19, 6, 7, 24, 18, 11, 28, 23, 14, 22, 5, 36, ...
17, 14, 22, 19, 18, 8, 20, 10, 23, 15, 4, 1, 3, 24, 13, 16, 33, 9, ...
18, 16, 23, 24, 25, 26, 14, 13, 17, 19, 22, 9, 5, 6, 8, 10, 15, 27, ...
...
Look at the entry in the second cell in row 3. It can't be a 1, because the 1 in cell(1,2) is a knight's move away, it can't be a 2, 3, 4, or 5, because it is adjacent to cells containing these numbers, and there is a 6 in cell (1,3) that is a knight's move away. The smallest free number is therefore 7.
MAPLE
# Based on Alois P. Heinz's program for A269526
A:= proc(n, k) option remember; local m, s;
if n=1 and k=1 then 1
else s:= {seq(A(i, k), i=1..n-1),
seq(A(n, j), j=1..k-1),
seq(A(n-t, k-t), t=1..min(n, k)-1),
seq(A(n+j, k-j), j=1..k-1)};
# add knights moves
if n >= 3 then s:={op(s), A(n-2, k+1)}; fi;
if n >= 3 and k >= 2 then s:={op(s), A(n-2, k-1)}; fi;
if n >= 2 and k >= 3 then s:={op(s), A(n-1, k-2)}; fi;
if k >= 3 then s:={op(s), A(n+1, k-2)}; fi;
for m while m in s do od; m
fi
end:
[seq(seq(A(1+d-k, k), k=1..d), d=1..15)];
MATHEMATICA
A[n_, k_] := A[n, k] = Module[{m, s}, If[n==1 && k==1, 1, s = Join[Table[ A[i, k], {i, 1, n-1}], Table[A[n, j], {j, 1, k-1}], Table[A[n-t, k-t], {t, 1, Min[n, k]-1}], Table[A[n+j, k-j], {j, 1, k-1}]] // Union; If[n >= 3, AppendTo[s, A[n-2, k+1]] // Union ]; If[n >= 3 && k >= 2, AppendTo[s, A[n-2, k-1]] // Union]; If[n >= 2 && k >= 3, AppendTo[s, A[n-1, k-2]] // Union]; If[k >= 3, AppendTo[s, A[n+1, k-2]] // Union]; For[m = 1, MemberQ[s, m], m++]; m]]; Table[A[1+d-k, k], {d, 1, 15}, {k, 1, d}] // Flatten (* Jean-François Alcover, Mar 14 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane following a suggestion from Joseph G. Rosenstein, Jul 07 2016
STATUS
approved