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Palindromes themselves are not 'Reverse and Add!'ed, so they yield a zero!

Numbers n that may have a(n) = -1 (i.e., potential Lychrel numbers) appear in A023108. - Michael De Vlieger, Jan 11 2018

Record indices and values are given in A065198 and A065199. - M. F. Hasler, Feb 16 2020

The starting number n is regarded as part of the trajectory, so palindromes are excluded from the sequence. A088753 is obtained if palindromes are not excluded. The smallest term in A063048 but not in A088753 is 19098, the smallest term in A088753 but not in A063048 is 9999.

Subsequence of A023108. Sequence A070788 is similarly defined, but palindromes are irrelevant. Corresponding sequences for other bases are A075252 (base 2), A077405 (base 3), A075421 (base 4).

If the trajectory of a number k joins the trajectory of a smaller number which is a term of the present sequence, then this occurs after very few Reverse and Add! steps (at most 8 for k < 100000, at most 10 for k < 1000000). On the other hand, the trajectories of the terms < 14000 do not join the trajectory of any smaller term within at least 1500 steps. This is the precise meaning of "presumably" in the definition.

The terms are rather unevenly distributed. They form clusters, especially above 10^4, 10^5, 10^6, ... . The interval from 10000 to 11000 for example contains 26 terms, whereas only two terms occur in the interval from 90000 to 100000.

It seems that if the most significant digit is not equal to 1, the least significant digit is always 9, while this does not hold for the Lychrel numbers as in A023108. - A.H.M. Smeets, Feb 18 2019

From A.H.M. Smeets, Sep 18 2021: (Start)

Let d_0 d_1 d_2 ... d_n be the decimal digits of an (n+1)-digit number.

All numbers in this sequence seem to satisfy the following condition:

d_0 = "1" or d_n = "9", and for all k, 0 < k < floor((n-1)/2), d_k = "0" or d_k = "9" or d_(n-k) = "0" or d_(n-k) = "9".

The plot log_10(a(n)) versus log_10(n) shows a stepwise behavior. However, the global behavior seems to be a straight line with slope e/(e-1) (= A185393). This slope is also obtained for the seeds in the Reverse and Add! problem in other bases. (End)

According to J. Walker, Ronald Sprague has proved that this trajectory does not contain a palindrome. [I would like a reference for this.] Another proof has been given by Klaus Brockhaus.

10110 is the smallest number with this property in base 2. The analogous number in base 10 is believed to be 196, but its trajectory (see A006960) has never been proved not to contain a palindrome.

The binary numbers have a regular pattern with cycle length 4:

a(4k) = 101^(k+1)010^(k+1) for k >= 1,

a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,

a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,

a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation. - A.H.M. Smeets, Feb 03 2019

From A.H.M. Smeets, Feb 11 2019: (Start)

Pattern with cycle length 4 represented by contextfree grammars with production rules:

S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;

S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;

S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;

S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;

see also A058042 for similar grammars for the binary represented trajectory of 77. (End)

Normally one stops as soon as a palindrome is reached.

A Reverse and Add! sequence.

Trajectories of 25, 34, 43, 52, 59, 61, 68, 70, 86, 95, ..., merge into this sequence. - Robert G. Wilson v, Dec 16 2005

The conjecture that the trajectories of the terms of this sequence do not join is based on the observation that if the trajectories of two integers below 10000 join, this happens after at most 15 steps, while for any two terms the trajectories do not join within 1200 steps. For pairs from 1, 3, 5, 7, 9, 100, 102, 106 this has even been checked for 10000 steps.

The positive integers are the domain of the equivalence relation 'the trajectories of a and b join'; each of its presumably infinitely many equivalence classes is represented by a term of this sequence. Each class contains infinitely many integers (cf. A070789 - A070798). In such a class the relation 'the trajectory of a is part of the trajectory of b' is a partial order for which a term c is a maximal element if it is in A067031 (integers not of the form k + reverse(k) for any k) and the integer at which the trajectories of a and b join is the greatest lower bound of a and b.

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A063048, i.e. whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e. a(n) = -1 for n > 12.

Subsequence of A023108.

Since we do not even know if 196 eventually converges (see A006960, A023108) for n >= 3 these values are only conjectures.

A first 'Reverse and Add' operation is always made, even if the starting value n is already a palindrome, in contrast to the variant A033665.

It is conjectured that a(196) = -1, see A023108.

Because A061563 has offset 0 one should add a(0) = 1 here. - Wolfdieter Lang, Jan 12 2018

Record indices and values beyond a(1) = 1 and a(5) = 2 are given in A065198 and A065199: These refer to the variant A033665 (main entry with more up-to-date references), as can be seen from A065199(1..3) = (0, 1, 2) for A065198(1..3) = (0, 10, 19). But all larger records correspond to a non-palindromic index n, in which case a(n) = A033665(n). - M. F. Hasler, Feb 16 2020

Normally one stops as soon as a palindrome is reached.

A Reverse and Add! sequence.