A246141 Index sequence for limit-block extending A006337 (difference sequence of the Beatty sequence for sqrt(2)) with first term as initial block.

1, 3, 6, 8, 15, 20, 27, 32, 37, 49, 66, 78, 90, 107, 119, 136, 148, 160, 177, 189, 206, 235, 247, 276, 305, 317, 346, 375, 404, 416, 445, 474, 486, 515, 556, 585, 614, 655, 684, 725, 754, 783, 824, 853, 894, 923, 964, 993, 1022, 1063, 1092, 1133, 1162, 1191

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 A006337 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 A246140 and A246141.

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

G. C. Greubel, Table of n, a(n) for n = 1..550

Example

S = A006337, with B = (s(1)); that is, (m,k) = (1,0)
S = (1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2,...)
B'(0) = (1)
B'(1) = (1,2)
B'(2) = (1,2,1)
B'(3) = (1,2,1,1)
B'(4) = (1,2,1,1,2)
B'(5) = (1,2,1,1,2,1)
S^ = (1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1,...),
with index sequence (1,3,6,8,15,...)

Mathematica

seqPosition1[list_, seqtofind_] := If[Length[#] > Length[list], {}, Last[Last[      Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 1]]]] &[seqtofind]; s =  Differences[Table[Floor[n*Sqrt[2]], {n, 10000}]]; 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] (* A246140 *)
Accumulate[Table[p[k], {k, 0, n - 1}]] (* A246141 *)

Crossrefs

Cf. A246140, A246127, A246142, A246144, A246147, A006337.

Sequence in context: A213983 A174133 A261928 * A051212 A143869 A165298

Adjacent sequences: A246138 A246139 A246140 * A246142 A246143 A246144

Keyword

nonn

Author

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

Status

approved