

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.


4



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
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OFFSET

1,2


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 largest number in row n is 2n.


LINKS



EXAMPLE

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;
...


MATHEMATICA

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]]]


CROSSREFS

This is a base2 analog of A323460.


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



