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 A274630 Square array T(n,k) (n>=1, k>=1) read by antidiagonals upwards in which the number entered in a square is the smallest positive number that is different from the numbers already filled in that are queens' or knights' moves away from that square. 7
 1, 2, 3, 4, 5, 6, 3, 7, 8, 2, 5, 1, 9, 4, 7, 6, 2, 10, 11, 1, 5, 7, 4, 12, 6, 3, 9, 8, 8, 9, 11, 13, 2, 10, 6, 4, 10, 12, 1, 3, 4, 7, 13, 11, 9, 9, 6, 2, 5, 8, 1, 12, 14, 3, 10, 11, 13, 3, 7, 6, 14, 9, 5, 1, 12, 15, 12, 8, 4, 14, 9, 11, 10, 3, 15, 2, 7, 13, 13, 10, 5, 1, 12, 15, 2, 16, 6, 4, 8, 14, 11 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A269526, A000027. For first column, row, and main diagonal see A274631, A274632, A274633. See A308883 for position of 1 in column n. See A308201 for an essentially identical array. Sequence in context: A195153 A328764 A323081 * A278058 A320115 A319994 Adjacent sequences:  A274627 A274628 A274629 * A274631 A274632 A274633 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane following a suggestion from Joseph G. Rosenstein, Jul 07 2016 STATUS approved

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Last modified December 13 09:54 EST 2019. Contains 329968 sequences. (Running on oeis4.)