[ *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(3r-1)/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 n-gonal is **r/2[(****sides**–2)r–(**sides**–4)]

Jeff Heleen re-discovered 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.