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A304453 An expanded binary notation for n: the normal binary expansion for n is expanded by mapping each 1 to 10 and retaining the existing 0's.

%I A304453 #16 Jan 08 2021 16:52:55
%S A304453 0,10,100,1010,1000,10010,10100,101010,10000,100010,100100,1001010,
%T A304453 101000,1010010,1010100,10101010,100000,1000010,1000100,10001010,
%U A304453 1001000,10010010,10010100,100101010,1010000,10100010,10100100,101001010,10101000,101010010,101010100,1010101010,1000000,10000010,10000100
%N A304453 An expanded binary notation for n: the normal binary expansion for n is expanded by mapping each 1 to 10 and retaining the existing 0's.
%C A304453 This notation is used by Penrose for specifying Turing machine examples. In general, this notation is much more compact than unary. Because no number encoded by this notation contains two or more consecutive 1's, there are an infinite number of additional strings available such as 110, 1110, 11110, ... for specifying delimiters (in lists of numbers), operations, etc. In some contexts, zero may alternately be represented by no symbol at all, for example, when there are two immediately-consecutive delimiters (commas).
%D A304453 R. Penrose, The Emperor's New Mind, Oxford, 1989, pp. 42-46.
%H A304453 Rick L. Shepherd, <a href="/A304453/b304453.txt">Table of n, a(n) for n = 0..9999</a>
%e A304453 a(3) = 1010 because 3 in binary is A007088(3) = 11 and each 1 has been replaced by 10 here. Similarly, a(4) = 1000 because A007088(4) = 100 and the expansion adds another 0 after the 1.
%p A304453 a:= n-> (l-> parse(cat(seq(10*l[-i], i=1..nops(l)))))(convert(n, base, 2)):
%p A304453 seq(a(n), n=0..42);  # _Alois P. Heinz_, Jan 08 2021
%t A304453 Table[FromDigits[Flatten[IntegerDigits[n,2]/.(1->{1,0})]],{n,0,40}] (* _Harvey P. Dale_, Sep 07 2019 *)
%o A304453 (PARI)
%o A304453 {a(n) = my(B, k);
%o A304453         if(n >= 0,
%o A304453           B = List(binary(n)); k = 1;
%o A304453           while(k <= #B,
%o A304453             if(B[k] == 1,
%o A304453               k++; listinsert(B, 0, k));
%o A304453             k++);
%o A304453           sum(k = 1, #B, B[k]*(10^(#B - k))))}
%o A304453 (Python)
%o A304453 def a(n): return int(bin(n)[2:].replace('1', '10'))
%o A304453 print([a(n) for n in range(35)]) # _Michael S. Branicky_, Jan 08 2021
%Y A304453 Cf. A007088.
%K A304453 nonn,base
%O A304453 0,2
%A A304453 _Rick L. Shepherd_, May 12 2018