A323456 -id:A323456 - cp's OEIS frontend

A323455 Irregular triangle read by rows: row n lists the numbers that can be obtained from the binary expansion of n by inserting a single 0 after any 1.

2, 4, 5, 6, 8, 9, 10, 10, 12, 11, 13, 14, 16, 17, 18, 18, 20, 19, 21, 22, 20, 24, 21, 25, 26, 22, 26, 28, 23, 27, 29, 30, 32, 33, 34, 34, 36, 35, 37, 38, 36, 40, 37, 41, 42, 38, 42, 44, 39, 43, 45, 46, 40, 48, 41, 49, 50, 42, 50, 52, 43, 51, 53, 54, 44, 52, 56

Offset: 1

N. J. A. Sloane, Jan 16 2019

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

			From 7 = 111 we can get 1011 = 11, 1101 = 13, and 1110 = 14, so row 7 is {11,13,14}.
The triangle begins:
  2,
  4,
  5, 6,
  8,
  9, 10,
  10, 12,
  11, 13, 14,
  16,
  17, 18,
  18, 20,
  19, 21, 22,
  ...

Michael S. Branicky, Table of n, a(n) for n = 1..10870 (Rows 1..2000, flattened)

Cf. A000120. See A323456 for a closely related sequence, the binary analog of A323386.

r323455[n_] := Module[{digs=IntegerDigits[n, 2]}, Map[FromDigits[#, 2]&, Map[Insert[digs, 0, #+1]&, Flatten[Position[digs, 1]]]]] (* nth row *)
a323455[{m_, n_}] := Flatten[Map[r323455, Range[m, n]]]
a323455[{1, 28}] (* Hartmut F. W. Hoft, Oct 24 2023 *)

def row(n):
    b = bin(n)[2:]
    s = set(b[:i+1] + "0" + b[i+1:] for i in range(len(b)) if b[i] == "1")
    return sorted(int(w, 2) for w in s)
print([c for n in range(1, 29) for c in row(n)]) # Michael S. Branicky, Jul 24 2022

More terms from David Consiglio, Jr., Jan 17 2019

a(49) and beyond from Michael S. Branicky, Jul 24 2022

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A323455 Irregular triangle read by rows: row n lists the numbers that can be obtained from the binary expansion of n by inserting a single 0 after any 1.

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