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A246147 Index sequence for limit-block extending A010060 (Thue-Morse sequence) with first term as initial block. 4
0, 3, 6, 12, 20, 30, 36, 68, 92, 116, 132, 156, 180, 228, 260, 308, 356, 420, 452, 516, 564, 612, 676, 708, 756, 804, 836, 900, 948, 996, 1076, 1188, 1268, 1316, 1460, 1572, 1716, 1764, 1844, 1956, 2100, 2212, 2292, 2340, 2484, 2740, 2868, 3060, 3252, 3380 (list; graph; refs; listen; history; text; internal format)
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 A010060 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 A246147.

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=0..49.

EXAMPLE

S = A010060, with B = (s(0)); that is, (m,k) = (0,0)

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

B'(0) = (0)

B'(1) = (0,1)

B'(2) = (0,1,1)

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

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

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

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

with index sequence (0,3,6,12,20,30,36,68,...)

MATHEMATICA

seqPosition1[list_, seqtofind_] := If[Length[#] > Length[list], {}, Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 1]]]] &[seqtofind]; s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 14]; (* A010060 *)

Take[s, 60]

t = {{0}}; 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] (* A246146 *)

-1 + Accumulate[Table[p[k], {k, 0, n - 1}]] (* A246147 *)

CROSSREFS

Cf. A246146, A246127, A246140, A246142, A246144, A010060.

Sequence in context: A320192 A160732 A320608 * A066140 A200067 A061061

Adjacent sequences:  A246144 A246145 A246146 * A246148 A246149 A246150

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified July 24 20:26 EDT 2021. Contains 346273 sequences. (Running on oeis4.)