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The successive numbers of digits between two "0"s in the sequence are given by the sequence itself.
1

%I #5 May 23 2016 22:07:09

%S 1,0,100,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,

%T 27,28,29,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,

%U 53,54,55,56,57,58,59,110,120,61,10,62,130,63,64,20,65,66,140,67,68,69,30,71,72,73,150,74,75,76,77,40,78,79,81,82,83,50,84,85,86,87,88,160,89,91,92,93,94,95,60,96,97,98,99,111,1110,112,113,114,115,1120,116,117,118,119,121,70,122,123,124,125,126,170,127,128,129,131,132,1130,133,134,135,136,137,138,80,139,141,142,143,144,145,1140,146,147,148,149,151,152,153,90,154,155,156,157

%N The successive numbers of digits between two "0"s in the sequence are given by the sequence itself.

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

%e There is 1 digit between the first two "0" of the sequence [1,0,100]; then there are 0 digits between the next pair of "0": [100]; then there are 100 digits between the next pair of "0": [100,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,110]; then there are 2 digits between the next pair of "0": [110,120], etc. The quantities 1, 0, 100, 2,... rebuild the starting sequence. The next "gaps" will thus be of 3 digits, then 4, then 5...

%K nonn,base

%O 1,3

%A _Eric Angelini_ and _Lars Blomberg_, May 23 2016