A389336 Rectangular array, (C(n,k)), read by descending antidiagonals: C(n,k) = n++k, the concatenation sum of n and k, defined as the index of the concatenation w(n)w(k) in the sequence S = (w(n)) of all 01-words in lexicographic order.

3, 4, 5, 7, 6, 7, 8, 11, 8, 9, 9, 12, 15, 10, 11, 10, 13, 16, 19, 12, 13, 15, 14, 17, 20, 23, 14, 15, 16, 23, 18, 21, 24, 27, 16, 17, 17, 24, 31, 22, 25, 28, 31, 18, 19, 18, 25, 32, 39, 26, 29, 32, 35, 20, 21, 19, 26, 33, 40, 47, 30, 33, 36, 39, 22, 23, 20

Offset

1,1

Comments

Regarding the indexing of the sequence S of all 01-words by positive integers, see A390512 and A390513. The indexing begins with w(1) = 0, w(2) = 1, w(3) = 00, w(4) = 01, w(5) = 10, w(6) = 11.

Links

Table of n, a(n) for n=1..67.

Example

C(6,4) = 6++4 = 28 = the index of 1101 (the concatenation of 11 and 01).
Corner:
   3    4    7    8    9   10   15   16   17   18   19   20
   5    6   11   12   13   14   23   24   25   26   27   28
   7    8   15   16   17   18   31   32   33   34   35   36
   9   10   19   20   21   22   39   40   41   42   43   44
  11   12   23   24   25   26   47   48   49   50   51   52
  13   14   27   28   29   30   55   56   57   58   59   60
  15   16   31   32   33   34   63   64   65   66   67   68
  17   18   35   36   37   38   71   72   73   74   75   76
  19   20   39   40   41   42   79   80   81   82   83   84
  21   22   43   44   45   46   87   88   89   90   91   92
  23   24   47   48   49   50   95   96   97   98   99  100

Mathematica

u[n_] := u[n] = Map[StringJoin[Map[ToString, Rest[IntegerDigits[#, 2]]]] &, Range[2, 2^(n + 1) - 1]];
t = u[10]; s[n_, k_] := StringJoin[t[[n]], t[[k]]];
v = Map[Flatten, Table[Position[Map[# == s[n, k] &, t], True], {n, 1, 20}, {k, 1, 20}]];
Grid[v]   (* A389336, array *)
u[n_, k_] := v[[n]][[k]]; Table[u[n - k + 1, k], {n, 20}, {k, n, 1, -1}] // Flatten (* A389336, sequence *)

Crossrefs

Cf. A005408 (column 1), A005843 (column 2), A020330, A059939 (number of occurrences of each positive integer), A092754 (row 1), A206332 (row 2), A390512, A390513.

Sequence in context: A202702 A388178 A380187 * A389466 A049465 A196122

Adjacent sequences: A389333 A389334 A389335 * A389337 A389338 A389339

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 07 2025

Status

approved