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A288936
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Fixed point of the mapping 00->0010, 01->011, 10->011, starting with 00.
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3
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0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1
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OFFSET
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1
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COMMENTS
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Conjecture: the number of letters (0's and 1's) in the n-th iterate of the mapping is given by A000124.
Proof of the conjecture: let N0(n) be the number of 0's, and N1(n) be the number of 1's in the n-th iterate of the StringReplace mapping defining (a(n)).
CLAIM: N0(n) = n+2, N1(n) = n(n+1)/2 for all n=1,2,...
The proof is by induction. We have N0(1) =3, N1(1) = 1. The crucial observation is that 00 only occurs at position 1 and position 4, and that at position 4 it always is preceded by 1. Moreover, each 0 (except the first one), combines either as 01 or as 10, which both give 011 in the next iterate.
So N0(n+1) = N0(n) + 1 = n+3, and
N1(n+1) = N1(n) + N0(n) - 1 = n(n+1)/2 + n+1 = (n+1)(n+2)/2.
This proves the conjecture: N0(n)+N1(n) = (n^2 +3n+4)/2 = A000124(n+1) for n=1,2,.... (End)
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LINKS
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FORMULA
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The first six iterates of the mapping:
00
0010
0010011
00100110111
0010011011011111
0010011011011101111111
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MATHEMATICA
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z = 11; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "011", "10" -> "011"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48 (* A288936 *)
Flatten[Position[st, 0]] (* A288937 *)
Flatten[Position[st, 1]] (* A288938 *)
Table[StringLength[w[n]], {n, 0, 20}] (* A000124 conjectured *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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