Binary Strings With Integers Not Occurring Earlier

[Leroy Quet]

I find the following sequences (below) that I just submitted to be 
interesting. But their definitions are a little long and unwieldy, for my 
taste. (Maybe someone could improve the definitions, as well as comment, 
calculate more terms, check the terms I already submitted.)
And even though these sequences are new as far as the OEIS is concerned, 
they are not original. Where can I find (preferably on-line) info 
regarding similar topics?

And what are the 3 decimal sequences asymptotic to? Is there anything 
else interesting that seq.fan users can add?

thanks,
Leroy Quet

>%S A118247 
>0,1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,1,0,1,1,
>0,1,1,0,0,1,1,1,1,0,1,1,1,1,1,1
>%N A118247 a(0)=0. Concatenate onto the end of the sequence (from left to 
>right) the integer m_n converted into binary (with the most significant 
>digit on the left), where m_n is the lowest integer > A118248(n-1) and 
>whose binary representation does not occur anywhere earlier in the 
>sequence (when the concatenated sequence is read from left to right). 
>A118248(n) then equals m_n when written in decimal.
>%e A118247 The sequence begins 0,1,1,0,1,0,0,1,1,1,1,0,0,0. Now A118248(5) 
>= 8, which is represented by the 1,0,0,0 at the end of the sequence. The 
>binary representation of 9 (1001 in binary) and (decimal) 10 (1010 in 
>binary) both occur earlier in the sequence. But the binary representation 
>of (decimal) 11 (1011 in binary) does not occur earlier in the sequence, 
>so (1,0,1,1) is added to the end of the sequence. And A118248(6) becomes 11.
>%Y A118247 A118248,A118249,A118251
>%O A118247 0
>%K A118247 ,easy,more,nonn,

>%S A118248 0,1,2,4,7,8,11,16,18,21,22,25,29,31
>%N A118248 Described at sequence A118247.
>%Y A118248 A118247,A118250,A118252
>%O A118248 0
>%K A118248 ,easy,more,nonn,

>%S A118249 
>0,1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,1,
>1,1,0,1
>%N A118249 a(0)=0. Concatenate onto the end of the sequence (from left to 
>right) the integer m_n converted into binary and reversed (with the most 
>significant digit on the right), where m_n is the lowest integer > 
>A118250(n-1) and whose reversed binary representation does not occur 
>anywhere earlier in the sequence (when the concatenated sequence is read 
>from left to right). A118250(n) then equals m_n when written in decimal.
>%e A118249 The sequence begins 0,1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0,1. Now 
>A118250(6) = 10 (decimal), which is represented by the 0,1,0,1 at the end 
>of the sequence. The binary representation of (decimal) 11 (1101 in binary 
>and reversed) and 12 (0011 in binary and reversed) both occur earlier in 
>the sequence. But the binary representation of 13 (1011 in binary and 
>reversed) does not occur earlier in the sequence, so (1,0,1,1) is added to 
>the end of the sequence. And A118250(7) becomes 13.
>%Y A118249 A118247,A118250,A118251
>%O A118249 0
>%K A118249 ,easy,more,nonn,

>%S A118250 0,1,3,4,5,8,10,13,15,16,18,23
>%N A118250 Described at sequence A118249.
>%Y A118250 A118248,A118249,A118252
>%O A118250 0
>%K A118250 ,easy,more,nonn,

>%S A118251 
>1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,1,1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,1,1,0,
>1,0,0,0,1,1
>%N A118251 a(1)=1. Concatenate onto the end of the sequence (from left to 
>right) the integer m_n converted into binary and reversed (with the most 
>significant digit on the right), where m_n is the lowest integer > 
>A118252(n-1) and whose reversed binary representation does not occur 
>anywhere earlier in the sequence (when the concatenated sequence is read 
>from left to right). A118252(n) then equals m_n when written in decimal.
>%e A118251 The sequence begins 
>1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,1,1,1,0,1,0,0,1,1. Now A118252(8) = 12, 
>which is represented by the 0,0,1,1 at the end of the sequence. The binary 
>representation of 13 (1011 in binary and reversed) and 14 (0111 in binary 
>and reversed) both occur earlier in the sequence. But the binary 
>representation of 15 (1111 in binary and reversed) does not occur earlier 
>in the sequence, so (1,1,1,1) is added to the end of the sequence. And 
>A118252(9) becomes 15.
>%Y A118251 A118247,A118249,A118252
>%O A118251 1
>%K A118251 ,easy,more,nonn,

>%S A118252 1,2,3,4,8,10,11,12,15,16,22,24
>%N A118252 Described at sequence A118251.
>%Y A118252 A118248,A118250,A118251
>%O A118252 1
>%K A118252 ,easy,more,nonn,