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

0,2

Comments

First column is A083051, main diagonal is A083052, antidiagonal sums give A083053.

A083050 is the dispersion of the sequence given by floor(1+n*sqrt(2)); for a discussion of dispersions, see A191429.

Links

Table of n, a(n) for n=0..68.

Example

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 ...

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 *)

Crossrefs

Cf. A083051, A083052, A083053, A083044, A083047, A191429.

Sequence in context: A216623 A297551 A297673 * A194030 A353658 A083044

Adjacent sequences: A083047 A083048 A083049 * A083051 A083052 A083053

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 18 2003

Status

approved