A323465 Irregular triangle read by rows: row n lists the numbers that can be obtained from the binary expansion of n by either deleting a single 0, or inserting a single 0 after any 1, or doing nothing.

1, 2, 1, 2, 4, 3, 5, 6, 2, 4, 8, 3, 5, 9, 10, 3, 6, 10, 12, 7, 11, 13, 14, 4, 8, 16, 5, 9, 17, 18, 5, 6, 10, 18, 20, 7, 11, 19, 21, 22, 6, 12, 20, 24, 7, 13, 21, 25, 26, 7, 14, 22, 26, 28, 15, 23, 27, 29, 30, 8, 16, 32, 9, 17, 33, 34, 9, 10, 18, 34, 36, 11, 19

Offset

1,2

Comments

All the numbers in row n have the same binary weight (A000120) as n.

If k appears in row n, n appears in row k.

If we form a graph on the positive integers by joining k to n if k appears in row n, then there is a connected component for each weight 1, 2, ...

The largest number in row n is 2n.

The smallest number in the component containing n is 2^A000120(n)-1, and n is reachable from 2^A000120(n)-1 in A023416(n) steps. - Rémy Sigrist, Jan 26 2019

Links

Rémy Sigrist, Table of n, a(n) for n = 1..8208

Example

From 6 = 110 we can get 6 = 110, 11 = 3, 1010 = 10, or 1100 = 12, so row 6 is {3,6,10,12}.
From 7 = 111 we can get 7 = 111, 1011 = 11, 1101 = 13, or 1110 = 14, so row 7 is {7,11,13,14}.
The triangle begins:
   1,  2;
   1,  2,  4;
   3,  5,  6;
   2,  4,  8;
   3,  5,  9, 10;
   3,  6, 10, 12;
   7, 11, 13, 14;
   4,  8, 16;
   5,  9, 17, 18;
   5,  6, 10, 18, 20;
   7, 11, 19, 21, 22;
   6, 12, 20, 24;
   7, 13, 21, 25, 26;
   7, 14, 22, 26, 28;
  15, 23, 27, 29, 30;
   8, 16, 32;
  ...

Mathematica

r323465[n_] := Module[{digs=IntegerDigits[n, 2]} , Map[FromDigits[#, 2]&, Union[Map[Insert[digs, 0, #+1]&, Flatten[Position[digs, 1]]], Map[Drop[digs, {#}]&, Flatten[Position[digs, 0]]], {digs}]]] (* nth row *)
a323465[{m_, n_}] := Flatten[Map[r323465, Range[m, n]]]
a323465[{1, 22}] (* Hartmut F. W. Hoft, Oct 24 2023 *)

Crossrefs

Cf. A000120, A323286, A323455, A323456, A323466 (number of terms in each row), A323467 (minimal number in each row).

This is a base-2 analog of A323460.

Sequence in context: A302655 A316997 A377274 * A364780 A124904 A187500

Adjacent sequences: A323462 A323463 A323464 * A323466 A323467 A323468

Keyword

nonn,tabf,look,base

Author

N. J. A. Sloane, Jan 26 2019

Extensions

More terms from Rémy Sigrist, Jan 27 2019

Status

approved