A277861 - OEIS

%I #23 Mar 30 2017 23:38:29

%S 1,2,3,4,5,6,7,8,9,10,11,12,20,22,23,24,25,26,27,28,29,30,13,32,33,34,

%T 35,36,37,38,39,40,14,15,16,42,43,44,45,46,47,48,49,50,17,18,19,21,52,

%U 53,54,55,56,57,58,59,60,62,63,64,65,66,67,68,69,70,72

%N A self-describing sequence: the zeros and ones in the decimal representation of the sequence correspond to the binary representation of the sequence.

%C The sequence is always extended with the smallest integer not yet present and not leading to a contradiction.

%C Numbers with neither 0 nor 1 in their decimal representation appear in increasing order.

%C Comments from _N. J. A. Sloane_, Nov 10 2016: (Start)

%C Here is another way to state the condition that the sequence must satisfy.

%C For each n, let D denote the concatenation of the first n terms in base 10, and let B denote the concatenation of the first n terms in base 2. Let D' be obtained from D by deleting all digits except 0 and 1.

%C Then D' must be a prefix of B.

%C In the examples below, D' appears in parentheses.

%C A278030 shows indices where the sequence changes from on or below the diagonal (a(n)<=n) to above the diagonal (a(n)>n), or vice versa. (End)

%H Rémy Sigrist, <a href="/A277861/b277861.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A277861/a277861.pl.txt">PERL program for A277861</a>

%e When computing the sequence, we must check that for any n > 0, the zeros and ones among the decimal representation of the first n terms match the beginning of the binary representation of these terms.

%e The following table depicts the first terms, alongside their binary representation, and the matching zeros and ones among their decimal representation (in parentheses):

%e n   a(n)  a(n) in binary  First n terms in binary

%e --  ----  --------------  -----------------------

%e 1   1     1               (1)

%e 2   2     10              (1)10

%e 3   3     11              (1)1011

%e 4   4     100             (1)1011100

%e 5   5     101             (1)1011100101

%e 6   6     110             (1)1011100101110

%e 7   7     111             (1)1011100101110111

%e 8   8     1000            (1)10111001011101111000

%e 9   9     1001            (1)101110010111011110001001

%e 10  10    1010            (110)11100101110111100010011010

%e 11  11    1011            (11011)1001011101111000100110101011

%e 12  12    1100            (110111)0010111011110001001101010111100

%e 13  20    10100           (1101110)01011101111000100110101011110010100

%Y Cf. A007088, A278030.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Nov 02 2016