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A274528 Square array read by antidiagonals upwards: T(n,k) = A269526(n+1,k+1) - 1, n>=0, k>=0.

%I A274528 #88 Mar 07 2020 13:50:20
%S A274528 0,1,2,2,3,1,3,0,4,5,4,1,5,0,3,5,6,2,1,7,4,6,7,0,4,8,2,9,7,4,8,3,0,6,
%T A274528 5,10,8,5,3,6,1,7,4,11,12,9,10,6,2,4,5,8,3,13,7,10,11,7,8,5,9,2,6,14,
%U A274528 15,13,11,8,12,9,10,13,3,14,15,16,6,17,12,9,13,10,2,3,7,15,8,5,11,14,6
%N A274528 Square array read by antidiagonals upwards: T(n,k) = A269526(n+1,k+1) - 1, n>=0, k>=0.
%C A274528 This sequence has essentially the same properties as the main sequence A269526, but now involves the nonnegative integers.
%C A274528 This version is important because of the following comment from _Allan C. Wechsler_, originally contributed to A269526. - _N. J. A. Sloane_, Jun 30 2016
%C A274528 Sprague-Grundy (Nim) values for a combinatorial game played with two piles of counters. Legal moves consist of removing any positive number of counters from either pile, or removing the same number from both piles, or moving any positive number of counters from the right pile to the left pile. If the Nim-values (as in Sprague-Grundy theory) are written in an array indexed by the number of counters in the two piles, we obtain this array. - _Allan C. Wechsler_, Jun 29 2016 [corrected by _N. J. A. Sloane_, Sep 25 2016]
%C A274528 The same sequence arises if we construct a triangle, by reading from left to right in each row, always choosing the smallest nonnegative number which does not produce a duplicate number in any row or diagonal. - _N. J. A. Sloane_, Jul 02 2016
%C A274528 It appears that the numbers generally appear for the first time in or near the first few rows. - _Omar E. Pol_, Jul 03 2016
%H A274528 Alois P. Heinz, <a href="/A274528/b274528.txt">Antidiagonals n = 0..200, flattened</a>
%H A274528 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A274528 Rémy Sigrist, <a href="/A274528/a274528.png">Colored illustration of T(n, k) for n = 0..499 and k = 0..499</a> (where the color is function of T(n, k))
%H A274528 N. J. A. Sloane, <a href="/A195264/a195264.pdf">Confessions of a Sequence Addict (AofA2017)</a>, slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
%e A274528 The corner of the square array begins:
%e A274528 0,  2,  1,  5,  3,  4,  9, 10, 12,  7, 13, 17,
%e A274528 1,  3,  4,  0,  7,  2,  5, 11, 13, 15,  6,
%e A274528 2,  0,  5,  1,  8,  6,  4,  3, 14, 16,
%e A274528 3,  1,  2,  4,  0,  7,  8,  6, 15,
%e A274528 4,  6,  0,  3,  1,  5,  2, 14,
%e A274528 5,  7,  8,  6,  4,  9,  3,
%e A274528 6,  4,  3,  2,  5, 13,
%e A274528 7,  5,  6,  8, 10,
%e A274528 8, 10,  7,  9,
%e A274528 9, 11, 12,
%e A274528 10, 8,
%e A274528 11,
%p A274528 # From _N. J. A. Sloane_, Jul 30 2018, based on Heinz's program in A269526
%p A274528 A:= proc(n, k) option remember; local m, s;
%p A274528          if n=1 and k=1 then 0
%p A274528        else s:= {seq(A(i, k), i=1..n-1),
%p A274528                  seq(A(n, j), j=1..k-1),
%p A274528                  seq(A(n-t, k-t), t=1..min(n, k)-1),
%p A274528                  seq(A(n+j, k-j), j=1..k-1)};
%p A274528             for m from 0 while m in s do od; m
%p A274528          fi
%p A274528      end:
%p A274528 [seq(seq(A(1+d-k, k), k=1..d), d=1..12)];
%t A274528 A[n_, k_] := A[n, k] = Module[{m, s}, If[n==1 && k==1, 0, 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}]]; For[m = 0, MemberQ[s, m], m++]; m]];
%t A274528 Table[A[d-k+1, k], {d, 1, 13}, {k, 1, d}] // Flatten (* _Jean-François Alcover_, May 03 2019, from Maple *)
%Y A274528 Cf. A269526, A274529, A274534, A274650, A274652.
%Y A274528 Columns 1, 2, 3, 4 give A001477, A004443, A274615, A274619.
%K A274528 nonn,tabl,easy,look
%O A274528 0,3
%A A274528 _Omar E. Pol_, Jun 29 2016