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.

22, 77, 442, 537, 775, 1066, 1081, 1082, 1085, 1115, 1562, 1575, 1587, 2173, 3355, 3599, 3871, 4099, 4153, 4185, 4193, 4202, 4262, 4285, 4402, 4633, 4666, 6163, 6166, 6374, 9241, 9466, 16544, 16546, 16586, 16601, 16613, 16616, 16720, 16748, 16994

Offset: 1

Author

Klaus Brockhaus, Sep 10 2002

Keywords

• base
• nonn

Comments

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.

From A.H.M. Smeets, Feb 12 2019: (Start)

Most terms in this sequence eventually give rise to a regular binary pattern. These regular patterns can be represented by contextfree grammars:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | A_a(n);

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | B_a(n);

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | C_a(n) and

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | D_a(n).

A_22 = 1101, B_22 = 1000, C_22 = 1101, D_22 = 0010 (see also A058042);

A_77 = 1100010, B_77 = 0000101, C_77 = 1101011, D_77 = 0100000 (see also A075253)

Decimal representations for 10 A_a(n) 00 are given by A306514(n).

Binary representations for 10 A_a(n) 00 are given by A306515(n).

(End)

Examples

			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.
		

Links

• Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
• Index entries for sequences related to Reverse and Add!

Crossrefs

Cf. A063048, A075421, A066059, A058042, A061561, A075253, A306514, A306515.

Programs

Mathematica

limit = 10^2; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
utraj = {};
Select[Range[0, 17000], (x = NestWhileList[# + IntegerReverse[#, 2] &, #, # != IntegerReverse[#, 2] & , 1, limit];
   If[Length[x] >= limit  && Intersection[x, utraj] == {},
    utraj = Union[utraj, x]; True,
utraj = Union[utraj, x]]) &] (* Robert Price, Oct 16 2019 *)