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 A246142 Limiting block extension of A004539 (base-2 representation of sqrt(2)) with first term as initial block. 7
 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 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 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 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, 10000000][]; Take[s, 60] t = {{1}}; p = seqPosition1[s, Last[t]]; s = Drop[s, p]; 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. A246143, A246128, A246145, A246147, A004539. Sequence in context: A128810 A123272 A266623 * A091219 A304109 A275601 Adjacent sequences:  A246139 A246140 A246141 * A246143 A246144 A246145 KEYWORD nonn,base,more AUTHOR Clark Kimberling and Peter J. C. Moses, Aug 17 2014 STATUS approved

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Last modified August 3 14:40 EDT 2021. Contains 346438 sequences. (Running on oeis4.)