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

Author

N. J. A. Sloane, Jan 16 2019

Keywords

• nonn
• base
• tabf

Comments

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

Examples

			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,
  ...
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

Python

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

Extensions

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

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