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The successive numbers of digits between two "3"s in the sequence are given by the sequence itself.
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%I #5 May 23 2016 22:02:09

%S 0,1,2,3,30,31,4,32,5,6,34,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,

%T 24,35,25,26,27,28,29,40,41,42,44,45,46,47,48,49,50,36,51,13,52,54,55,

%U 56,57,58,59,60,61,62,64,65,66,67,68,69,37,70,71,38,72,74,23,75,76,77,78,79,80,81,82,84,85,86,87,88,89,90,91,92,39,94,95,96,300,97,98,99,301,100,101,43,102,104,105,53,106,107,108,103,109,110,111,112,302,114,115,116,117,304,118,119,120,121,63,122,124,125,126,127,73,128,129,140,141,142,113,144,145,146,147,148,149,305,150

%N The successive numbers of digits between two "3"s in the sequence are given by the sequence itself.

%C This is the lexicographically first such sequence, starting with a(1)=0, that has always been extended with the smallest integer not already present in the sequence and not leading to a contradiction.

%e There are 0 digits between the first two "3" of the sequence [0,1,2,3,30]; then there is 1 digit between the next pair of "3": [30,31]; then there are 2 digits between the next pair of "3": [31,4,32]; then there are 3 digits between the next pair of "3": [32,5,6,34]; then there are 30 digits between the next pair of "3" [34,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,24,35], etc. The quantities 0, 1, 2, 3, 30,... rebuild the starting sequence.

%K nonn,base

%O 1,3

%A _Eric Angelini_ and _Lars Blomberg_, May 23 2016