A323465 - OEIS

%I #26 Nov 22 2023 22:23:42

%S 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,

%T 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,

%U 23,27,29,30,8,16,32,9,17,33,34,9,10,18,34,36,11,19

%N 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.

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

%C If k appears in row n, n appears in row k.

%C 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, ...

%C The largest number in row n is 2n.

%C 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 26 2019

%H Rémy Sigrist, <a href="/A323465/b323465.txt">Table of n, a(n) for n = 1..8208</a>

%e From 6 = 110 we can get 6 = 110, 11 = 3, 1010 = 10, or 1100 = 12, so row 6 is {3,6,10,12}.

%e From 7 = 111 we can get 7 = 111, 1011 = 11, 1101 = 13, or 1110 = 14, so row 7 is {7,11,13,14}.

%e The triangle begins:

%e    1,  2;

%e    1,  2,  4;

%e    3,  5,  6;

%e    2,  4,  8;

%e    3,  5,  9, 10;

%e    3,  6, 10, 12;

%e    7, 11, 13, 14;

%e    4,  8, 16;

%e    5,  9, 17, 18;

%e    5,  6, 10, 18, 20;

%e    7, 11, 19, 21, 22;

%e    6, 12, 20, 24;

%e    7, 13, 21, 25, 26;

%e    7, 14, 22, 26, 28;

%e   15, 23, 27, 29, 30;

%e    8, 16, 32;

%e   ...

%t 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 *)

%t a323465[{m_, n_}] := Flatten[Map[r323465, Range[m, n]]]

%t a323465[{1, 22}] (* _Hartmut F. W. Hoft_, Oct 24 2023 *)

%Y Cf. A000120, A323286, A323455, A323456, A323466 (number of terms in each row), A323467 (minimal number in each row).

%Y This is a base-2 analog of A323460.

%K nonn,tabf,look,base

%O 1,2

%A _N. J. A. Sloane_, Jan 26 2019

%E More terms from _Rémy Sigrist_, Jan 27 2019