

A241422


Limitreverse of the infinite Fibonacci word A003849 with first term as initial block.


4



0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0
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OFFSET

0


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 A003849 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*. The index sequence for A241422 is A245921. Indeed, the sequence S* for the classical Fibonacci word A003849 is essentially that of the (2,1)version, as indicated in the Formula section.


LINKS

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


FORMULA

a(n) = A245920(n)  d(n), where d(n) = 2 if n is even and d(n) = 0 if n is odd.


EXAMPLE

S = the infinite Fibonacci word A003849, with B = (s(0)); that is, (m,k) = (0,0)
S = (0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,...);
B'(0) = (0)
B'(1) = (0,1)
B'(2) = (0,1,0)
B'(3) = (0,1,0,1,0)
B'(4) = (0,1,0,1,0,0)
B'(5) = (0,1,0,1,0,0,1)
S* = (0,1,0,1,0,0,1,0,1,0,0,1,0,0,1,0,...),
with index sequence (0,2,5,7,15,...) = A245921.


MATHEMATICA

z = 100; seqPosition2[list_, seqtofind_] := Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 2]]] &[seqtofind]; n = 18; s = Flatten[Nest[{#, #[[1]]} &, {0, 1}, n]] (* A003849 *);
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. A245920, A003849, A245921.
Sequence in context: A135993 A285966 A215530 * A189661 A145573 A092202
Adjacent sequences: A241419 A241420 A241421 * A241423 A241424 A241425


KEYWORD

nonn,obsc


AUTHOR

Clark Kimberling and Peter J. C. Moses, Aug 08 2014


STATUS

approved



