A390513 Rectangular array, (U(n,k)), read by descending antidiagonals. Let S be the sequence of all 01 words in lexicographic order. Then U(n,k) is the position in A390512 of the index of the word 0^(k-1)w(n), where w(n) is the n-th word in S that is 0 or has first letter 1.

1, 3, 2, 7, 4, 5, 15, 8, 9, 6, 31, 16, 17, 10, 11, 63, 32, 33, 18, 19, 12, 127, 64, 65, 34, 35, 20, 13, 255, 128, 129, 66, 67, 36, 21, 14, 511, 256, 257, 130, 131, 68, 37, 22, 23, 1023, 512, 513, 258, 259, 132, 69, 38, 39, 24

Offset

1,2

Comments

The array (U(n,k)) shows the indices in A390512 of the array of all 01 words represented by this corner:

0 00 000 0000 00000 000000 0000000

1 01 001 0001 00001 000001 0000001

10 010 0010 00010 000010 0000010 00000010

11 011 0011 00011 000011 0000011 00000011

100 0100 00100 000100 0000100 00000100 000000100

Column 1 of (U(n,k)) is A206332, so that (U(n,k)) is the dispersion of the complement of A206332.

Links

Table of n, a(n) for n=1..55.

Example

Corner of (U(n,k)):
    1     3     7     15     31     63     127     255
    2     4     8     16     32     64     128     256
    5     9    17     33     65    129     257     513
    6    10    18     34     66    130     258     514
   11    19    35     67    131    259     515    1027
   12    20    36     68    132    260     516    1028
   13    21    37     69    133    261     517    1029
   14    22    38     70    134    262     518    1030
   23    39    71    135    263    519    1031    2055
   24    40    72    136    264    520    1032    2056

Mathematica

r = 9; r1 = 9; (* r = #rows of T, r1= #rows to show *)
c = 10; c1 = 10; (* c = #cols of T, c1 = #cols to show *)
u = Table[n - 1 + 2^Floor[1 + Log[2, n]], {n, 1, 20000}];
v = Rest[Complement[Range[Max[u]], u]];  (* Will form dispersion of v *)
f[n_] := v[[n]];
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, Floor[N[Log[2, Length[v]]]]]};
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}]]    (* array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]  (* sequence *)

Crossrefs

Cf. A000027, A000051 (row 3), A000079 (row 2), A000225 (row 1), A001057, A052548 (row 4), A206332 (column 1), A345254, A390512.

Sequence in context: A375890 A370698 A303763 * A303765 A255555 A191664

Adjacent sequences: A390510 A390511 A390512 * A390514 A390515 A390516

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 18 2025

Status

approved