[ 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."
Klaus Brockhaus OEIS sequences
A068065, A068064, A068062, A068061, and A067030.
A000127 