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.
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, 26, 21, 25, 39, 46, 47, 42, 37, 30, 24, 36, 56, 66, 67, 60, 53, 43, 34, 28, 51, 80, 94, 95, 85, 75, 61, 49, 40, 31, 73, 114, 133, 135, 121, 107, 87, 70, 57, 44, 35, 104, 162, 189
Offset: 0
Examples
Table begins: 1 2 3 5 8 12 17 25 36 51 73 104 ... 4 6 9 13 19 27 39 56 80 114 162 230 ... 7 10 15 22 32 46 66 94 133 189 268 380 ... 11 16 23 33 47 67 95 135 191 271 384 544 ... 14 20 29 42 60 85 121 172 244 346 490 693 ... 18 26 37 53 75 107 152 215 305 432 611 865 ... 21 30 43 61 87 124 176 249 353 500 708 1002 ... 24 34 49 70 99 141 200 283 401 568 804 1138 ... 28 40 57 81 115 163 231 327 463 655 927 1311 ... 31 44 63 90 128 182 258 365 517 732 1036 1466 ... 35 50 71 101 143 203 288 408 577 817 1156 1635 ...
Programs
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Mathematica
(* program generates the dispersion array T of the complement of increasing sequence f[n] *) r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *) c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *) x = Sqrt[2]; f[n_] := Floor[n*x + 1] (* f(n) is complement of column 1 *) mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]] rows = {NestList[f, 1, c]}; Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A083050 as an array *) Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *) (* Program by Peter J. C. Moses, Jun 01 2011, added here Jun 03 2011 by Clark Kimberling *)
Comments