|
%I #10 Jul 27 2018 09:36:34
%S 1,3,6,8,15,20,27,32,37,49,66,78,90,107,119,136,148,160,177,189,206,
%T 235,247,276,305,317,346,375,404,416,445,474,486,515,556,585,614,655,
%U 684,725,754,783,824,853,894,923,964,993,1022,1063,1092,1133,1162,1191
%N Index sequence for limit-block extending A006337 (difference sequence of the Beatty sequence for sqrt(2)) with first term as initial block.
%C 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).
%C 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.
%C 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).
%H G. C. Greubel, <a href="/A246141/b246141.txt">Table of n, a(n) for n = 1..550</a>
%e S = A006337, with B = (s(1)); that is, (m,k) = (1,0)
%e S = (1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2,...)
%e B'(0) = (1)
%e B'(1) = (1,2)
%e B'(2) = (1,2,1)
%e B'(3) = (1,2,1,1)
%e B'(4) = (1,2,1,1,2)
%e B'(5) = (1,2,1,1,2,1)
%e S^ = (1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1,...),
%e with index sequence (1,3,6,8,15,...)
%t 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 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 *)
%t Accumulate[Table[p[k], {k, 0, n - 1}]] (* A246141 *)
%Y Cf. A246140, A246127, A246142, A246144, A246147, A006337.
%K nonn
%O 1,2
%A _Clark Kimberling_ and _Peter J. C. Moses_, Aug 17 2014
|
