A323460 -id:A323460 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 26 2019

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

			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;
  ...

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

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.

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 *)

More terms from Rémy Sigrist, Jan 27 2019

A358708 Starting from 1, successively take the smallest "Choix de Bruxelles" (A323286) which is not already in the sequence.

Original entry on oeis.org

1, 2, 4, 8, 16, 13, 23, 26, 46, 43, 83, 86, 166, 133, 136, 68, 34, 17, 27, 47, 87, 167, 137, 174, 172, 171, 271, 272, 236, 118, 19, 29, 49, 89, 169, 139, 178, 278, 239, 269, 469, 439, 478, 474, 237, 267, 467, 437, 837, 867, 1667, 1337, 1367, 687, 347, 177, 277, 477, 877, 1677, 1377, 1747, 1727, 1717, 1734, 1732, 866, 433, 233, 263, 163, 323, 313, 316, 38, 76, 73, 143, 123, 63, 33, 36, 18, 9

Offset: 0

Alon Vinkler, Nov 26 2022

The Choix de Bruxelles doubles or halves some decimal digit substring and rows of A323286 are all ways this can be done.
So a(n) is the smallest term of the row a(n-1) of A323286 which is not among {a(0..n-1)}.
The sequence is finite since having reached 18 -> 9 the sole Choix for 9 would be back to 18, which is already in the sequence.

			Below, square brackets [] represent multiplication by 2 (e.g., [6] = 12); curly brackets {} represent division by 2 (e.g., {6} = 3); digits outside the brackets are not affected by the multiplication or division (e.g., 1[6] = 112 and 1{14} = 17).
We begin with 1 and, at each step, we go to the smallest number possible that hasn't yet appeared in the sequence:
 1 --> [1]  =  2
 2 --> [2]  =  4
 4 --> [4]  =  8
 8 --> [8]  = 16
 16 --> 1{6} = 13
 13 --> [1]3 = 23
 23 --> 2[3] = 26
 26 --> [2]6 = 46
 ... and so on.

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444 [math.NT], Feb 2019; Fib. Quart. 57:3 (2019), 195-200.
Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, "Choix de Bruxelles": A New Operation on Positive Integers, Local copy.
Alon Vinkler, C# Program

Cf. A323460, A307635, A323286, A323454.

A323464 Values of n at which A323484 reaches a new record.

Original entry on oeis.org

1, 2, 3, 6, 20, 40, 80, 1975, 3999, 15995, 39999

Offset: 1

N. J. A. Sloane, Jan 23 2019

The corresponding numbers of steps are 0, 1, 11, 12, 13, 14, 15, 16, 17.

			a(4)=20 refers to the fact that it takes 13 steps to reach 100 from 5 using the Choix de Bruxelles (version 2) operation, and all multiples of 5 less than 100 can be reached from 5 in fewer than 13 steps.

Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane, "Choix de Bruxelles": A New Operation on Positive Integers, arXiv:1902.01444, Feb 2019; Fib. Quart. 57:3 (2019), 195-200.
Eric Angelini, Lars Blomberg, Charlie Neder, Remy Sigrist, and N. J. A. Sloane,, "Choix de Bruxelles": A New Operation on Positive Integers, Local copy.

Cf. A323286, A323460, A323454, A323484.

a(10) from Michael S. Branicky, Oct 05 2024

a(11) from Michael S. Branicky, Oct 11 2024

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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.

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A358708 Starting from 1, successively take the smallest "Choix de Bruxelles" (A323286) which is not already in the sequence.

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A323464 Values of n at which A323484 reaches a new record.

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