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A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

%I A075252 #26 Nov 03 2019 22:41:29
%S A075252 22,77,442,537,775,1066,1081,1082,1085,1115,1562,1575,1587,2173,3355,
%T A075252 3599,3871,4099,4153,4185,4193,4202,4262,4285,4402,4633,4666,6163,
%U A075252 6166,6374,9241,9466,16544,16546,16586,16601,16613,16616,16720,16748,16994
%N A075252 Trajectory of n under the Reverse and Add! operation carried out in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.
%C A075252 Base-2 analog of A063048 (base 10) and A075421 (base 4); subsequence of A066059. - For the trajectory of 22 (cf. A061561) and the trajectory of 77 (cf. A075253) it has been proved that they do not contain a palindrome. A similar proof can be given for most terms of this sequence, but there are a few terms (4262, 17498, 33378, 33898, ...) whose trajectory does not show the kind of regularity that can be utilized for the construction of a proof. - If the trajectory of an integer k joins the trajectory of a smaller integer which is a term of the present sequence, then this occurs after very few 'Reverse and Add!' steps (at most 84 for numbers < 20000). On the other hand, the trajectories of the terms of this sequence do not join the trajectory of any smaller term within at least 1000 steps.
%C A075252 From _A.H.M. Smeets_, Feb 12 2019: (Start)
%C A075252 Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:
%C A075252 S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);
%C A075252 S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);
%C A075252 S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and
%C A075252 S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).
%C A075252 A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);
%C A075252 A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)
%C A075252 Decimal representations for 10 A_a(n) 00 are given by A306514(n).
%C A075252 Binary representations for 10 A_a(n) 00 are given by A306515(n).
%C A075252 (End)
%H A075252 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A075252 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%e A075252 442 is a term since the trajectory of 442 (presumably) does not lead to an integer which occurs in the trajectory of 22 or of 77.
%t A075252 limit = 10^2; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
%t A075252 utraj = {};
%t A075252 Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # != IntegerReverse[#, 2] & , 1, limit];
%t A075252    If[Length[x] >= limit  && Intersection[x, utraj] == {},
%t A075252     utraj = Union[utraj, x]; True,
%t A075252 utraj = Union[utraj, x]]) &] (* _Robert Price_, Oct 16 2019 *)
%Y A075252 Cf. A063048, A075421, A066059, A058042, A061561, A075253, A306514, A306515.
%K A075252 base,nonn
%O A075252 1,1
%A A075252 _Klaus Brockhaus_, Sep 10 2002