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

%I A370698 #24 Mar 12 2024 22:29:46
%S A370698 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,
%T A370698 41,74,85,255,16,19,26,43,82,149,170,511,24,23,34,45,84,165,298,341,
%U A370698 1023,28,25,36,53,86,169,330,597,682,2047,30,27,38,69,90,171
%N A370698 Rectangular array, read by antidiagonals: row n consists of the numbers m whose binary representation has exactly n runs.
%C A370698 Every positive integer occurs exactly once, and for every n, the numbers in row n have the parity of n.
%e A370698 Corner:
%e A370698     1    3    7   15   31   63  127  255
%e A370698     2    4    6    8   12   14   16   24
%e A370698     5    9   11   13   17   19   23   25
%e A370698    10   18   20   22   26   34   36   38
%e A370698    21   37   41   43   45   53   69   73
%e A370698    42   74   82   84   86   90  106  138
%e A370698    85  149  165  169  171  173  181  213
%e A370698   170  298  330  338  340  342  346  362
%e A370698   341  597  661  677  681  683  685  693
%e A370698 The binary representation of 22 is 10110, which has 4 runs: 1, 0, 11, 0.
%t A370698 a[n_] := a[n] = Select[Range[8000], Length[Split[IntegerDigits[#, 2]]] == n &];
%t A370698 t[n_, k_] := a[n][[k]];
%t A370698 Grid[Table[t[n, k], {n, 1, 12}, {k, 1, 12}]] (* array *)
%t A370698 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%Y A370698 Cf. A007089, A005811 (# runs in binary n), A000225 (row 1), A043569 (row 2), A043570 (row 3), A000975 (column 1), A370893 (ternary).
%K A370698 nonn,base,tabl
%O A370698 1,2
%A A370698 _Clark Kimberling_, Mar 11 2024