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A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.

%I A283734 #8 Mar 19 2017 01:13:40
%S A283734 1,2,3,4,5,7,6,8,10,12,9,11,14,16,19,13,15,18,21,24,28,17,20,23,26,30,
%T A283734 34,38,22,25,29,32,36,41,45,50,27,31,35,39,43,48,53,58,63,33,37,42,46,
%U A283734 51,56,61,67,72,78,40,44,49,54,59,65,70,76,82,88,95,47
%N A283734 Rank array, R, of the golden ratio, read by antidiagonals downwards.
%C A283734 Every row intersperses all other rows, and every column intersperses all other columns.  The array is the dispersion of the complement of column 1; column 1 is given by r(n) = r(n-1) + 1 + L(n), where L = lower Wythoff sequence (A000201).
%H A283734 Clark Kimberling, <a href="/A283734/b283734.txt">Antidiagonals n = 1..60, flattened</a>
%H A283734 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%F A283734 R(i,j) = R(i,0) + R(0,j) + i*j - 1, for i>=1, j>=1.
%e A283734 The corner of R begins:
%e A283734 1    2    4    6    9    13    17    22
%e A283734 3    5    8    11   15   20    25    31
%e A283734 7    10   14   18   23   29    35    42
%e A283734 12   16   21   26   32   39    46    54
%e A283734 19   24   30   36   43   51    59    68
%e A283734 28   34   41   48   56   65    74    84
%e A283734 38   45   53   61   70   80    90    101
%e A283734 50   58   67   76   86   97    108   120
%e A283734 Let t = golden ratio = (1 + sqrt(5))/2; then R(i,j) = rank of (j,i) when all nonnegative integer pairs (a,b) are ranked by the relation << defined as follows: (a,b) << (c,d) if a + b*t < c + d*t, and also (a,b) << (c,d) if a + b*t = c + d*t and b < d.  Thus R(2,1) = 10 is the rank of (1,2) in the list (0,0) << (1,0) << (0,1) << (2,0) << (1,1) << (3,0) << (0,2) << (2,1) << (4,0) << (1,2).
%t A283734 r = 40; r1 = 12;(*r=# rows of T,r1=# rows to show*);
%t A283734 c = 40; c1 = 12;(*c=# cols of T,c1=# cols to show*);
%t A283734 s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*GoldenRatio];
%t A283734 u = Table[s[n], {n, 0, 400}] (* A283733 *)
%t A283734 v = Complement[Range[Max[u]], u];
%t A283734 f[n_] := v[[n]]; Table[f[n], {n, 1, 30}]
%t A283734 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
%t A283734 Length[Union[list]]]; rows = {NestList[f, 1, c]};
%t A283734 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A283734 w[i_, j_] := rows[[i, j]];
%t A283734 TableForm[Table[w[i, j], {i, 1, r1}, {j, 1, c1}]]   (* A283734, array *)
%t A283734 Flatten[Table[w[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A283734, sequence *)
%t A283734 TableForm[Table[w[i, 1] + w[1, j] + (i - 1)*(j - 1) - 1, {i, 1, r1}, {j, 1, c1}]] (* A283734, array, by formula *)
%Y A283734 Cf. A001622, A255977 (row 1), A283733 (column 1), A000201, A087465.
%K A283734 nonn,tabl,easy
%O A283734 1,2
%A A283734 _Clark Kimberling_, Mar 16 2017