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A083050 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), where x = sqrt(2), n>=0, k>=0.

%I A083050 #13 Dec 01 2016 01:35:50
%S A083050 1,2,4,3,6,7,5,9,10,11,8,13,15,16,14,12,19,22,23,20,18,17,27,32,33,29,
%T A083050 26,21,25,39,46,47,42,37,30,24,36,56,66,67,60,53,43,34,28,51,80,94,95,
%U A083050 85,75,61,49,40,31,73,114,133,135,121,107,87,70,57,44,35,104,162,189
%N A083050 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), where x = sqrt(2), n>=0, k>=0.
%C A083050 First column is A083051, main diagonal is A083052, antidiagonal sums give A083053.
%C A083050 A083050 is the dispersion of the sequence given by floor(1+n*sqrt(2)); for a discussion of dispersions, see A191429.
%e A083050 Table begins:
%e A083050    1  2  3   5   8  12  17  25  36  51   73  104 ...
%e A083050    4  6  9  13  19  27  39  56  80 114  162  230 ...
%e A083050    7 10 15  22  32  46  66  94 133 189  268  380 ...
%e A083050   11 16 23  33  47  67  95 135 191 271  384  544 ...
%e A083050   14 20 29  42  60  85 121 172 244 346  490  693 ...
%e A083050   18 26 37  53  75 107 152 215 305 432  611  865 ...
%e A083050   21 30 43  61  87 124 176 249 353 500  708 1002 ...
%e A083050   24 34 49  70  99 141 200 283 401 568  804 1138 ...
%e A083050   28 40 57  81 115 163 231 327 463 655  927 1311 ...
%e A083050   31 44 63  90 128 182 258 365 517 732 1036 1466 ...
%e A083050   35 50 71 101 143 203 288 408 577 817 1156 1635 ...
%t A083050 (* program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t A083050 r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
%t A083050 c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
%t A083050 x = Sqrt[2]; f[n_] := Floor[n*x + 1]
%t A083050 (* f(n) is complement of column 1 *)
%t A083050 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A083050 rows = {NestList[f, 1, c]};
%t A083050 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A083050 t[i_, j_] := rows[[i, j]];
%t A083050 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A083050 (* A083050 as an array *)
%t A083050 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
%t A083050 (* Program by _Peter J. C. Moses_, Jun 01 2011, added here Jun 03 2011 by _Clark Kimberling_ *)
%Y A083050 Cf. A083051, A083052, A083053, A083044, A083047, A191429.
%K A083050 nonn,tabl
%O A083050 0,2
%A A083050 _Paul D. Hanna_, Apr 18 2003