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A370698
Rectangular array, read by antidiagonals: row n consists of the numbers m whose binary representation has exactly n runs.
1
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.
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: A108644 A194011 A375890 * A303763 A303765 A255555
KEYWORD
nonn,base,tabl
AUTHOR
Clark Kimberling, Mar 11 2024
STATUS
approved