[ June 3, 2001 ]
Polygonals with palindromic sides, rank and number
A list provided by Jeff Heleen (email)
Jeff borrowed the terminology from Albert H. Beiler's book
"Recreations in the Theory of Numbers", chapter XVIII.
Rank is just the number of the term (ie, 1st, 2nd, 3rd, etc.) for a particular polygon
(triangular, square, pentagonal, etc.) and number is just the result of using
the formula for that polygon with a given number of sides.
Example for a pentagon, sides = 5, formula is r(3r1)/2, where r = rank.
So, the 4th pentagonal number (r = 4) is 22, which is palindromic.
I highlighted myself the entry with sides = 2002,
to inform that the last palindromic year of our lives is advancing...
Sides < 10001  Rank < 101  Number 
3  3  6 
4  3  9 
5  4  22 
77  4  454 
141  4  838 
151  4  898 
797  4  4774 
7997  4  47974 
7  5  55 
55  5  535 
535  5  5335 
5335  5  53335 
6  6  66 
9  6  111 
11  6  141 
44  6  636 
424  6  6336 
4224  6  63336 
4554  6  68286 
383  7  8008 
646  7  13531 
11  9  333 
252  9  9009 
414  9  14841 
3  11  66 
4  11  121 
22  11  1111 
111  11  6006 
121  11  6556 
202  11  11011 
2002  11  110011 
2992  11  164461 
4  22  484 
343  33  180081 
707  33  372273 
747  33  393393 
1111  33  585585 
5  44  2882 
7  44  4774 
9  44  6666 
11  44  8558 
6  55  5995 
3  77  3003 
272  77  790097 
222  88  842248 
171  99  819918 
The general formula for a ngonal is r/2[(sides–2)r–(sides–4)]
Jeff Heleen rediscovered this list from his archive.
He cannot remember if the above list is exhaustive or not.
Can someone write a program to check out if the list is complete ? Thanks in advance !
[ Same day ]
Jeff rewrote the program from scratch and the output turned out to be slightly longer than the original list. He believes the current list
is now closer to being exhaustive within the set ranges.
