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A246145 Index sequence for limit-block extending A000002 (Kolakoski sequence) with first term as initial block. 4
1, 4, 13, 16, 51, 78, 97, 124, 178, 247, 322, 402, 475, 578, 623, 746, 842, 1030, 1111, 1173, 1454, 1481, 2071, 2385, 2686, 4395, 5402, 5587, 5932, 6150, 6622, 6767, 7038, 7311, 7461, 10404, 10674, 12797, 18358, 20169, 20575, 21667, 23244, 25101, 26224 (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 A000002 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 A246144 and A246145.

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..45.

EXAMPLE

S = A000002, with B = (s(1)); that is, (m,k) = (1,0)

S = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,...)

B'(0) = (1)

B'(1) = (1,1)

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

B'(3) = (1,1,2,2)

B'(4) = (1,1,2,2,1)

B'(5) = (1,1,2,2,1,2)

S^ = (1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,...),

with index sequence (1,4,13,16,51,78,97,124,178,247,322,...)

MATHEMATICA

seqPosition1[list_, seqtofind_] := If[Length[#] > Length[list], {}, Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 1]]]] &[seqtofind]; n = 30; s = Prepend[Nest[Flatten[Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, n], 1]; (* A246144 *)

Take[s, 30]

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] (* A246144*)

Accumulate[Table[p[k], {k, 0, n - 1}]] (*A246145*)

CROSSREFS

Cf. A246140, A246142, A246144, A246146, A000002.

Sequence in context: A191135 A268524 A032824 * A107462 A190863 A022132

Adjacent sequences:  A246142 A246143 A246144 * A246146 A246147 A246148

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)