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A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

%I A075421 #25 Oct 18 2019 04:11:45
%S A075421 290,318,719,795,799,1210,3903,4199,4207,4219,4236,4278,4279,4294,
%T A075421 4326,4333,4334,4338,4402,4598,4662,4726,5046,5357,6157,6174,7246,
%U A075421 7247,7295,7407,7549,8063,8191,9211,12319,12431,12463,12539,15487,16519,16587
%N A075421 Trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.
%C A075421 For 318 (cf. A075153), 266718 (cf. A075466) and 270798 (cf. A075467) one can prove that the base 4 trajectory does not contain a palindrome. A proof for 290 (cf. A075299) has not been found up to now. 4398859679359 is another known candidate (obtained from a remark of _David J. Seal_, cf. Links) for a term whose trajectory is provably palindrome-free, but is not secured that it does not join the trajectory of some term m < n. - 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 28 for k < 20000). On the other hand, the trajectories of the terms listed above do not join the trajectory of any smaller term within at least 1000 steps.
%C A075421 Base-4 analog of A063048 (base 10) and A075252 (base 2); subsequence of A075420.
%C A075421 From _A.H.M. Smeets_, Mar 18 2019: (Start)
%C A075421 David J. Seal (see LINKS) observed a cyclic pattern (length 6) in the trajectories that can be represented by an extended right regular grammar with production rules:
%C A075421 S -> S_a | S_b | S_c | S_d | S_e | S_f,
%C A075421 S_a -> 1033202000232 T_a, T_a -> 222 T_a | 2302333113230
%C A075421 S_b -> 2022321332331 T_b, T_b -> 111 T_b | 1223001203131
%C A075421 S_c -> 10002003002212 T_c, T_c -> 222 T_c | 3221333101333
%C A075421 S_d -> 103312202321111 T_d, T_d -> 111 T_d | 1102023122000
%C A075421 S_e -> 110200123122222 T_e, T_e -> 222 T_e | 2231232001301
%C A075421 S_f -> 213301021321111 T_f, T_f -> 111 T_f | 1113213003312
%C A075421 Within the first 471 terms of this sequence we observed three trajectories with a cyclic pattern (length 6) that can be represented by a context-free grammar with production rules:
%C A075421 S -> S_a | S_b | S_c | S_d | S_e | S_f,
%C A075421 S_a -> 10 T_a 00, T_a -> 3 T_a 0 | T_a0,
%C A075421 S_b -> 11 T_b 01, T_b -> 0 T_b 3 | T_b0,
%C A075421 S_c -> 22 T_c 12, T_c -> 0 T_c 3 | T_c0,
%C A075421 S_d -> 10 T_d 000, T_d -> 3 T_d 0 | T_d0,
%C A075421 S_e -> 11 T_e 301, T_e -> 0 T_e 3 | T_e0,
%C A075421 S_f -> 22 T_f 312, T_f -> 0 T_f 3 | T_f0.
%C A075421 The terminating strings in these context-free grammars are given by:
%C A075421 n     2      359                371
%C A075421 a(n)  318    266718             270798
%C A075421 T_a0  33230  33230000001033230  3323001033230
%C A075421 T_b0  03123  03123010001103123  0312302103123
%C A075421 T_c0  01313  01313120002201313  0131320201313
%C A075421 T_d0  33323  33323000001033323  3332300103323
%C A075421 T_e0  03222  03222301001103222  0322201113222
%C A075421 T_f0  02111  02111312002202111  0211112222111
%C A075421 From the fact that both, right regular grammars and context-free grammars occur, we wonder if other trajectories can be represented by context-sensitive grammars as well, by which other trajectories can be proven never to end up in a palindromic string? (End)
%H A075421 A.H.M. Smeets, <a href="/A075421/b075421.txt">Table of n, a(n) for n = 1..471</a>
%H A075421 Klaus Brockhaus, <a href="/A075421/a075421.gif">Illustration: Distribution of terms below 2000000</a>
%H A075421 David J. Seal, <a href="http://www.mathpages.com/home/dseal.htm">Results</a>
%H A075421 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%e A075421 719 is a term since the trajectory of 719 (presumably) does not lead to an integer which occurs in the trajectory of 290 or of 318.
%t A075421 limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
%t A075421 utraj = {};
%t A075421 Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 4] &, #, # !=IntegerReverse[#, 4] & , 1, limit];
%t A075421    If[Length[x] >= limit  && Intersection[x, utraj] == {},
%t A075421     utraj = Union[utraj, x]; True,
%t A075421 utraj = Union[utraj, x]]) &] (* _Robert Price_, Oct 16 2019 *)
%Y A075421 Cf. A063048, A075252, A075420, A075299, A075153, A075466, A075467, A091675.
%K A075421 base,nonn
%O A075421 1,1
%A A075421 _Klaus Brockhaus_, Sep 18 2002, revised Jan 28 2004