Remembering Reverse and Add, Palindromes and Trajectories

In an earlier post I commented on the reverse and add operation on numbers that usually leads to a palindrome e.g. 13 --> 31+13 --> 44, 26 --> 62+26 --> 88, 102 --> 201+102 --> 303 etc. However, some numbers (as far as can be determined) do not lead to palindromes under this operation. The first such number is 196. Here is a list of some such numbers taken from the OEIS A063048 entry:

196, 879, 1997, 7059, 10553, 10563, 10577, 10583, 10585, 10638, 10663, 10668, 10697, 10715, 10728, 10735, 10746, 10748, 10783, 10785, 10787, 10788, 10877, 10883, 10963, 10965, 10969, 10977, 10983, 10985, 12797, 12898, 13097, 13197, 13694

I've highlighted 10563 because my number for the day of this entry (24522) is connected to this number because it lies on its trajectory under the reverse and add operation. The list of such numbers defines OEIS A063064 (integers n > 10563 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10563) and begins thus:

11553, 12543, 13533, 14097, 14523, 15087, 15513, 16077, 16503, 17067, 18057, 18597, 19047, 19587, 20562, 21552, 22542, 24096, 24522, 25086, 25512, 26076, 26502, 27066, 28056, 28596, 29046, 29586, 30561, 31551, 32541, 33531, 34095 

Hence the title of this post. I was reminded about the reverse and add operation and how it mostly results in palindromes except for certain special "seed" numbers (such as 10563) that create seemingly endless "trajectories". It turns out that 24522 lies on the trajectory of 10563.

Below is an excerpt from the WolframMathWorld about what it terms the 196-Algorithm:

Take any positive integer of two digits or more, reverse the digits, and add to the original number. This is the operation of the reverse-then-add sequence. Now repeat the procedure with the sum so obtained until a palindromic number is obtained. This procedure quickly produces palindromic numbers for most integers. For example, starting with the number 5280 produces the sequence 5280, 6105, 11121, 23232. The end results of applying the algorithm to 1, 2, 3,  4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 121, ... (OEIS A033865). The value for 89 is especially large, being 8813200023188. The first few numbers not known to produce palindromes, sometimes known as Lychrel numbers (Van Landingham), are 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, ... (OEIS A023108).