A245939 - OEIS

%I #11 Aug 22 2014 10:13:25

%S 0,3,5,12,20,36,60,92,108,132,156,172,204,228,300,356,396,420,492,516,

%T 556,612,676,804,900,996,1124,1188,1316,1412,1508,1572,1668,1764,1892,

%U 1956,2052,2148,2340,2532,2788,2916,3108,3300,3428,3684,3876,4068,4196

%N Index sequence for limit-reversing A010060; see Comments.

%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 A014675 is such a sequence.)  Let B = B(m,k) = (s(m-k), s(m-k+1),...,s(m)) be such a block, where m >= 0 and k >= 0.  Let m(1) be the least i > m such that (s(i-k),s(i-k+1),...,s(i)) = B(m,k), and put B(m(1),k+1) = (s(m(1)-k-1),s(m(1)-k),...,s(m(1))).  Let m(2) be the least i > m(1) such that (s(i-k-1),s(i-k),...,s(i)) = B(m(1),k+1), and put B(m(2),k+2) = (s(m(2)-k-2),s(m(2)-k-1),...,s(m(2))).  Continuing in this manner gives a sequence of blocks B(m(n),k+n).  Let B'(n) = reverse(B(m(n),k+n)), so that for n >= 1, B'(n) comes from B'(n-1) by suffixing a single term; thus the limit of B'(n) is defined; we call it the "limit-reverse of S with initial block B(m,k)", denoted by S*(m,k), or simply S*.

%C ...

%C The sequence (m(i)), where m(0) = 0, is the "index sequence for limit-reversing S with initial block B(m,k)" or simply the index sequence for S*, as in A245939.

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

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

%e B'(0) = (0)

%e B'(1) = (0,1)

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

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

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

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

%e S* = (0,1,0,0,1,1,0,0,1,0,1,1,...),

%e with index sequence (0,3,5,12,20,36,60,92,108,132,...)

%t z = 50; seqPosition2[list_, seqtofind_] := Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 2]]] &[seqtofind]; n = 18; s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, n]; ans = Join[{s[[p[0] = pos = seqPosition2[s, #] - 1]]}, #] &[{s[[1]]}]; cfs = Table[s = Drop[s, pos - 1]; ans = Join[{s[[p[n] = pos = seqPosition2[s, #] - 1]]}, #] &[ans], {n, z}]; q = -1+Accumulate[Join[{1}, Table[p[n], {n, 0, z}]]]  (* A245939 *)

%Y Cf. A010060, A245938, A245920.

%K nonn

%O 0,2

%A _Clark Kimberling_ and _Peter J. C. Moses_, Aug 07 2014