A370698 Rectangular array, read by antidiagonals: row n consists of the numbers m whose binary representation has exactly n runs.

1, 3, 2, 7, 4, 5, 15, 6, 9, 10, 31, 8, 11, 18, 21, 63, 12, 13, 20, 37, 42, 127, 14, 17, 22, 41, 74, 85, 255, 16, 19, 26, 43, 82, 149, 170, 511, 24, 23, 34, 45, 84, 165, 298, 341, 1023, 28, 25, 36, 53, 86, 169, 330, 597, 682, 2047, 30, 27, 38, 69, 90, 171

Offset

1,2

Comments

Every positive integer occurs exactly once, and for every n, the numbers in row n have the parity of n.

Links

Table of n, a(n) for n=1..62.

Example

Corner:
    1    3    7   15   31   63  127  255
    2    4    6    8   12   14   16   24
    5    9   11   13   17   19   23   25
   10   18   20   22   26   34   36   38
   21   37   41   43   45   53   69   73
   42   74   82   84   86   90  106  138
   85  149  165  169  171  173  181  213
  170  298  330  338  340  342  346  362
  341  597  661  677  681  683  685  693
The binary representation of 22 is 10110, which has 4 runs: 1, 0, 11, 0.

Mathematica

a[n_] := a[n] = Select[Range[8000], Length[Split[IntegerDigits[#, 2]]] == n &];
t[n_, k_] := a[n][[k]];
Grid[Table[t[n, k], {n, 1, 12}, {k, 1, 12}]] (* array *)
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)

Crossrefs

Cf. A007089, A005811 (# runs in binary n), A000225 (row 1), A043569 (row 2), A043570 (row 3), A000975 (column 1), A370893 (ternary).

Sequence in context: A277679 A108644 A194011 * A303763 A303765 A255555

Adjacent sequences: A370695 A370696 A370697 * A370699 A370700 A370701

Keyword

nonn,base,tabl

Author

Clark Kimberling, Mar 11 2024

Status

approved