

A246128


Index sequence for limitblock extending the (2,1)version of the infinite Fibonacci word A014675 with first term as initial block.


10



0, 2, 7, 10, 15, 23, 31, 36, 44, 49, 57, 70, 78, 91, 104, 112, 125, 138, 159, 193, 214, 248, 282, 303, 337, 371, 392, 426, 447, 481, 515, 536, 570, 591, 625, 659, 680, 714, 748, 803, 892, 981, 1036, 1125, 1180, 1269, 1358, 1413, 1502, 1557, 1646, 1735, 1790
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OFFSET

0,2


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(m), s(m+1),...s(m+k)) be such a block, where m >= 0 and k >= 0. Let m(1) be the least i > m such that (s(i), s(i+1),...,s(i+k)) = B(m,k), and put B(m(1),k+1) = (s(m(1)), s(m(1)+1),...s(m(1)+k+1)). Let m(2) be the least i > m(1) such that (s(i), s(i+1),...,s(i+k)) = B(m(1),k+1), and put B(m(2),k+2) = (s(m(2)), s(m(2)+1),...s(m(2)+k+2)). Continuing in this manner gives a sequence of blocks B'(n) = B(m(n),k+n), so that for n >= 0, B'(n+1) comes from B'(n) by suffixing a single term; thus the limit of B'(n) is defined; we call it the "limiting block extension of S with initial block B(m,k)", denoted by S^.
...
The sequence (m(i)), where m(0) = 0, is the "index sequence for limitblock extending S with initial block B(m,k)", as in A246127.


LINKS

Table of n, a(n) for n=0..52.


EXAMPLE

S = the infinite Fibonacci word A014675, with 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) = (2,2)
B'(2) = (2,2,1)
B'(3) = (2,2,1,2)
B'(4) = (2,2,1,2,1)
B'(5) = (2,2,1,2,1,2)
S^ = (2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2,...),
with index sequence (0,2,7,10,15,...)


MATHEMATICA

seqPosition1[list_, seqtofind_] := If[Length[#] > Length[list], {}, Last[Last[ Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 1]]]] &[seqtofind]; s = Differences[Table[Floor[n*GoldenRatio], {n, 10000}]]; t = {{2}}; p[0] = seqPosition1[s, Last[t]]; s = Drop[s, p[0]]; Off[Last::nolast]; n = 1; While[(p[n] = seqPosition1[s, Last[t]]) > 0, (AppendTo[t, Take[s, {#, # + Length[Last[t]]}]]; s = Drop[s, #]) &[p[n]]; n++]; On[Last::nolast]; t1 = Last[t] (*A246127*)
q = 1 + Accumulate[Table[p[k], {k, 0, n  1}]] (*A246128*)


CROSSREFS

Cf. A245921, A246127, A246129, A014675.
Sequence in context: A190375 A066097 A035336 * A226830 A059316 A295825
Adjacent sequences: A246125 A246126 A246127 * A246129 A246130 A246131


KEYWORD

nonn


AUTHOR

Clark Kimberling and Peter J. C. Moses, Aug 15 2014


STATUS

approved



