[ *February 25, 2002* ]

Power of Patterns.

Illustrated by Klaus Brockhaus (email).

The pattern represents the new sequence A068065: Palindromes n

for which there is a unique k such that n = k + reverse(k).

E.g. 10801 = 10800 + 00801

and for no other k we have 10801 = k + reverse(k).

The remarkable gap in the second column of the following

pattern arises because 121 = 47 + 74 but also 121 = 110 + 011.

0 101 10001 1000001 100000001 ... | 2 ooo 10201 1002001 100020001 ... | 4 141 10401 1004001 100040001 ... | 6 161 10601 1006001 100060001 ... | 8 181 10801 1008001 100080001 ... | 11 1001 100001 10000001 1000000001 ... |

Asking if the dots at the bottom of each column

indicate an infinite pattern, or just unexplored terrain,

Klaus responded

"I have not worked out a rigorous proof, that the sequence continues in

the way indicated by the dots, but informal considerations make it clear

that the uniqueness condition is very __restrictive__ (if n = a + b and a is

not palindromic or ends with 0, then n = b + a is a second, different

representation) so some case distinctions concerning the number of

digits will lead to the desired result that the pattern is infinite.

A purely computational exploration of the larger numbers is not

feasible because of the required time."

Sloane sequences

A068065, A068064, A068062, A068061, and A067030.