[ *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