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