

A245920


Limitreverse of the (2,1)version of the infinite Fibonacci word A014675 with first term as initial block.


23



2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2
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OFFSET

0,1


COMMENTS

Suppose S = (s(0), s(1), s(2),...) is an infinite sequence such that every finite block of consecutive terms occurs infinitely many times in S. (It is assumed that A014675 is such a sequence.) Let B = B(m,k) = (s(mk), s(mk+1),...,s(m)) be such a block, where m >= 0 and k >= 0. Let m(1) be the least i > m such that (s(ik), s(ik+1),...,s(i)) = B(m,k), and put B(m(1),k+1) = (s(m(1)k1), s(m(1)k),...,s(m(1))). Let m(2) be the least i > m(1) such that (s(ik1), s(ik),...,s(i)) = B(m(1),k+1), and put B(m(2),k+2) = (s(m(2)k2), s(m(2)k1),...,s(m(2))). Continuing in this manner gives a sequence of blocks B(m(n),k+n). Let B'(n) = reverse(B(m(n),k+n)), so that for n >= 1, B'(n) comes from B'(n1) by suffixing a single term; thus the limit of B'(n) is defined; we call it the "limitreverse of S with initial block B(m,k)", denoted by S*(m,k), or simply S*.
The sequence (m(i)), where m(0) = 0, is the "index sequence for limitreversing S with initial block B(m,k)" or simply the index sequence for S*, as in A245921.
For numbers represented by taking S and S* as continued fractions, see A245975 and A245976. If S is taken to be the classical (0,1)version of the infinite Fibonacci word, then S* is obtained from the present sequence by substituting 0 for 2 throughout, as in A241422.
The limitreverse, S*, is analogous to a limiting block extension, S^, defined at A246127. The essential difference is that S^ is formed by extending each new block one term to the right, whereas S* is formed by extending each new block one term to the left (and then reversing).


LINKS

Clark Kimberling, Table of n, a(n) for n = 0..300


EXAMPLE

S = infinite Fibonacci word A014675, B = (s(0)); that is, (m,k) = (0,0);
S = (2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,...)
B'(0) = (2)
B'(1) = (2,1)
B'(2) = (2,1,2)
B'(3) = (2,1,2,1)
B'(4) = (2,1,2,1,2)
B'(5) = (2,1,2,1,2,2)
S* = (2,1,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,...),
with index sequence (0,2,5,7,15,...)


MATHEMATICA

z = 100; seqPosition2[list_, seqtofind_] := Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 2]]] &[seqtofind]; x = GoldenRatio; s = Differences[Table[Floor[n*x], {n, 1, z^2}]] ; ans = Join[{s[[p[0] = pos = seqPosition2[s, #]  1]]}, #] &[{s[[1]]}]; cfs = Table[s = Drop[s, pos  1]; ans = Join[{s[[p[n] = pos = seqPosition2[s, #]  1]]}, #] &[ans], {n, z}]; rcf = Last[Map[Reverse, cfs]]


CROSSREFS

Cf. A245921, A245922, A003849, A014675, A245975, A245976.
Sequence in context: A023518 A022921 A080763 * A165413 A172155 A080573
Adjacent sequences: A245917 A245918 A245919 * A245921 A245922 A245923


KEYWORD

nonn


AUTHOR

Clark Kimberling and Peter J. C. Moses, Aug 07 2014


STATUS

approved



