A323456 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.

2, 1, 4, 5, 6, 2, 8, 3, 9, 10, 3, 10, 12, 11, 13, 14, 4, 16, 5, 17, 18, 5, 6, 18, 20, 7, 19, 21, 22, 6, 20, 24, 7, 21, 25, 26, 7, 22, 26, 28, 23, 27, 29, 30, 8, 32, 9, 33, 34, 9, 10, 34, 36, 11, 35, 37, 38, 10, 12, 36, 40, 11, 13, 37, 41, 42, 11, 14, 38, 42, 44

Offset: 1

Author

N. J. A. Sloane, Jan 17 2019

Keywords

• nonn
• tabf
• base

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 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 17 2019

Examples

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

Links

• Rémy Sigrist, Rows n = 1..1000, flattened

Crossrefs

Cf. A000120, A323455, A323465.

This is a base-2 analog of A323286.

Programs

Mathematica

r323456[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]]]]]] (* nth row *)
a323456[{m_, n_}] := Flatten[Map[r323456, Range[m, n]]]
a323456[{1,22}] (* Hartmut F. W. Hoft, Oct 24 2023 *)

PARI

row(n) = { my (r=Set(), w=0, s=0); while (n, my (v=1+valuation(n,2)); r = setunion(r, Set(n*2^(w+1)+s)); if (v>1, r = setunion(r, Set(n*2^(w-1)+s))); s += (n%(2^v))*2^w; w += v; n \= 2^v); r } \\ Rémy Sigrist, Jan 27 2019

Python

def row(n):
    b = bin(n)[2:]
    s1 = set(b[:i+1] + "0" + b[i+1:] for i in range(len(b)) if b[i] == "1")
    s2 = set(b[:i] + b[i+1:] for i in range(len(b)) if b[i] == "0")
    return sorted(int(w, 2) for w in s1 | s2)
print([c for n in range(1, 23) for c in row(n)]) # Michael S. Branicky, Jul 24 2022

Extensions

More terms from Rémy Sigrist, Jan 27 2019