[ September 2, 2001 ]
A magical blending of different concepts
in a most beautiful pattern
Terry Trotter (email) and pdg

We start with palindromes whose digits smoothly go down and up.
These palindromes are then written out as a product of two near repdigits
and unexpectedly their sum reveals the number of the beast in all his shapes.
Finally we halt with the last entry which is a pandigital palindrome !

All these concepts are blended together in the following beautiful finite
pattern from Terry Trotter, quite a beastly menagerie...

9 = 3 * 3 , Sum = 6
989 = 23 * 43 , Sum = 66
98789 = 223 * 443 , Sum = 666
9876789 = 2223 * 4443 , Sum = 6666
987656789 = 22223 * 44443 , Sum = 66666
98765456789 = 222223 * 444443 , Sum = 666666
9876543456789 = 2222223 * 4444443 , Sum = 6666666
987654323456789 = 22222223 * 44444443 , Sum = 66666666
98765432123456789 = 222222223 * 444444443 , Sum = 666666666
9876543210123456789 = 2222222223 * 4444444443 , Sum = 6666666666

In an effort to create an infinite pattern I (pdg) took the above palindromes p
and calculated their p^th palindrome (1 is the 1^st palindrome).

The 989^th palindrome is 89098
The 98789^th palindrome is 887909788
The 9876789^th palindrome is 8876790976788
etc.

The pattern becomes clear now from the second entry on. The p^th palindrome
is constructed with 88 at the outer ends and 909 in the middle.
The zones inbetween are identical to the middle zone of p itself (98_767_89) !

The 98_765434567_89^th palindrome is
88_765434567_909_765434567_88

The 98_7654321234567_89^th palindrome is
88_7654321234567_909_7654321234567_88

The pattern holds even when the zeros start appearing in the middle.
So we have e.g.

The 98_7654321000001234567_89^th palindrome is
88_7654321000001234567_909_7654321000001234567_88

There is a Java engine for calculating the above palindromes at
The N^th Positive Integer Palindrome Generator