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

1,1

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

Links

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

Example

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,
  ...

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 *)

Prog

(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

Crossrefs

Cf. A000120, A323455, A323465.

This is a base-2 analog of A323286.

Sequence in context: A326056 A365689 A074720 * A326058 A262586 A058359

Adjacent sequences: A323453 A323454 A323455 * A323457 A323458 A323459

Keyword

nonn,tabf,base

Author

N. J. A. Sloane, Jan 17 2019

Extensions

More terms from Rémy Sigrist, Jan 27 2019

Status

approved