I confirm that Beech's numbers are palintiples, I will show that they are not all of the palintiples. I will show that the palintiples do not form a regular language. And then I will prove that I have found all the palintiples, by describing the them with a generalized form of regular expression. The results become more interesting in other bases. We exclude numbers starting with zeroes.
T is the carry out of that position.
D' to refer to a digit D on the left side of a string with a corresponding digit D' on the right side of the string. This enables us to construct a finite state machine to recognize those palintiples. I have written a computer program that constructs finite state machines for recognizing palintiples for various bases and constants. I found that base 10 is actually an unusually boring base for this problem. If anyone investigates the general behavior, I would be most happy to hear about it.