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A246143 Index sequence for limit-block extending A004539 (base 2 representation of sqrt(2)) with first term as initial block. 3
1, 3, 17, 18, 35, 45, 239, 341, 470, 1180, 1230, 2205, 5318, 45652, 68042, 73350, 119458, 388804, 475300, 773496, 836779, 845397, 1133816, 2670010 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 A004539 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^  in case the initial block is s(0).

The sequence (m(i)), where m(0) = 0, is the "index sequence for limit-block extending S with initial block B(m,k)", as in A246128.  If the sequence S is given with offset 1, then the role played by s(0) in the above definitions is played by s(1) instead, as in the case of A246142 and A246143.

Limiting block extensions are analogous to limit-reverse sequences, S*, defined at A245920.  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

Table of n, a(n) for n=1..24.

EXAMPLE

S = A004539, with B = (s(1)); that is, (m,k) = (1,0)

S = (1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, ...)

B'(0) = (1)

B'(1) = (1,1)

B'(2) = (1,1,1)

B'(3) = (1,1,1,0)

B'(4) = (1,1,1,0,0)

B'(5) = (1,1,1,0,0,1)

S^ = (1,1,1,0,0,1,1,0,1,0,0,1,1,0,...),

with index sequence (1,3,17,18,35,45,239,341,...)

MATHEMATICA

seqPosition1[list_, seqtofind_] := If[Length[#] > Length[list], {}, Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 1]]]] &[seqtofind]; s = RealDigits[Sqrt[2], 2, 10000000][[1]]; Take[s, 60]

t = {{1}}; 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]; Last[t] (*A246142*)

Accumulate[Table[p[k], {k, 0, n - 1}]] (*A246143*)

CROSSREFS

Cf. A246142, A246127, A246144, A246146, A004539.

Sequence in context: A043066 A293764 A295395 * A101144 A038886 A019342

Adjacent sequences:  A246140 A246141 A246142 * A246144 A246145 A246146

KEYWORD

nonn,more

AUTHOR

Clark Kimberling and Peter J. C. Moses, Aug 17 2014

STATUS

approved

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Last modified January 19 17:45 EST 2019. Contains 319309 sequences. (Running on oeis4.)