A323465 -id:A323465 - cp's OEIS frontend

A323466 Number of terms in row n of A323465.

2, 3, 3, 3, 4, 4, 4, 3, 4, 5, 5, 4, 5, 5, 5, 3, 4, 5, 5, 5, 6, 6, 6, 4, 5, 6, 6, 5, 6, 6, 6, 3, 4, 5, 5, 5, 6, 6, 6, 5, 6, 7, 7, 6, 7, 7, 7, 4, 5, 6, 6, 6, 7, 7, 7, 5, 6, 7, 7, 6, 7, 7, 7, 3, 4, 5, 5, 5, 6, 6, 6, 5, 6, 7, 7, 6, 7, 7, 7, 5, 6, 7, 7, 7, 8, 8, 8

Offset: 1

N. J. A. Sloane, Jan 26 2019

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Cf. A323465.

(* Function r323465[ ] for n-th row is defined in A323465 *)
a323466[n_] := Length[r323465[n]]
Map[a323466, Range[87]] (* Hartmut F. W. Hoft, Oct 24 2023 *)

More terms from Rémy Sigrist, Jan 27 2019

A323467 Smallest number in row n of A323465.

Original entry on oeis.org

1, 1, 3, 2, 3, 3, 7, 4, 5, 5, 7, 6, 7, 7, 15, 8, 9, 9, 11, 10, 11, 11, 15, 12, 13, 13, 15, 14, 15, 15, 31, 16, 17, 17, 19, 18, 19, 19, 23, 20, 21, 21, 23, 22, 23, 23, 31, 24, 25, 25, 27, 26, 27, 27, 31, 28, 29, 29, 31, 30, 31, 31, 63, 32, 33, 33, 35, 34, 35

Offset: 1

N. J. A. Sloane, Jan 26 2019

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Cf. A323465.

More terms from Rémy Sigrist, Jan 27 2019

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jan 17 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 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

			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,
  ...

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

Cf. A000120, A323455, A323465.

This is a base-2 analog of A323286.

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 *)

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

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

More terms from Rémy Sigrist, Jan 27 2019

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A323466 Number of terms in row n of A323465.

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A323467 Smallest number in row n of A323465.

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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.

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