A245935 - OEIS

%I #12 Sep 26 2016 21:42:13

%S 2,2,3,5,7,5,7,5,5,12,12,5,12,12,12,17,12,17,12,12,17,12,17,12,17,41,

%T 29,41,29,29,41,29,41,29,41,29,29,41,29,41,29,29,41,29,41,29,41,29,29,

%U 41,29,41,29,29,70,70,29,70,70,70,29,70,70,29,70,70,70

%N First differences of A245934; see Comments.

%C It appears that every term is a term of A002965.  The sequence A245935 arises from A245933 and A245934, in which the limit-reverse of certain sequences is defined, as follows.  Suppose that 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-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*.  (Since Beatty sequences are usually written with offset 1, the above definition is adapted accordingly, so that s(n) = A006337(n+1) for n >= 0.)

%C ...

%C The sequence (m(i)), where m(0) = 1, is the "index sequence for limit-reversing S with initial block B(m,k)" or simply the index sequence for S*, as in A245934.

%e S = A006337 (the Beatty sequence of sqrt(2)), re-indexed to start with s(0) = 1, with B = (s(0)); that is, (m,k) = (0,0)

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

%e B'(0) = (1)

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

%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* = (1,2,1,1,2,1,2,1,1,2,1,2,1,2,1,1,2,1,...), with index sequence (1,3,5,8,13,20,25,32,37,...), with difference sequence (2,2,3,5,7,5,7,5,5,12,12,...).

%t z = 140; seqPosition2[list_, seqtofind_] := Last[Last[Position[Partition[list, Length[#], 1], Flatten[{___, #, ___}], 1, 2]]] &[seqtofind]; x = Sqrt[2]; s =  Differences[Table[Floor[n*x], {n, 1, z^2}]];  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 = Accumulate[Join[{1}, Table[p[n], {n, 0, z}]]]  (* A245934 *)

%t q1 = Differences[q]  (* A245935 *)

%Y Cf. A245934, A245933, A245935, A245922.

%K nonn

%O 1,1

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