Let's consider three types of numbers:

- triangles: a(a+1)/2
- squares: b
^{2} - centered hexagons: 3c(c+1) + 1

And two questions:

- Which numbers are both triangular and square, triangular and hexagonal, square and hexagonal?
- Which triangles divide into smaller triangles?

*one or more recurrent series of the type U(a,b,c)F*, all with the same factor.

__The first question__

can be generalized to diophantine equations of the type p

_{1}a

^{2}+q

_{1}a + r

_{1}= p

_{2}b

^{2}+q

_{2}b + r

_{2}, where p

_{i}, q

_{i}and r

_{i}are constants.

Here are the above mentioned examples:

__Triangles through squares: a(a+1)/2 = b__

^{2}a: | 1 | 8 | 49 | 288 | 1681 | 9800 | ... | U(0,1,2)6. |

b: | 1 | 6 | 35 | 204 | 1189 | 6930 | ... | U(0,1,0)6, or the primus of 6. |

__Triangles through hexagons: a(a+1)/2 = 3c(c+1)+1__

a: | 1 | 13 | 133 | 1321 | 13081 | 129493 | ... | U(1,1,4)10. |

c: | 0 | 5 | 54 | 539 | 5340 | 52865 | ... | U(-1,0,4)10. |

__Squares through hexagons: b__

^{2}= 3c(c+1)+1b: | 1 | 13 | 181 | 2521 | 35113 | 489061 | ... | U(1,1,0)14, or the tertius of 14. |

c: | 0 | 7 | 104 | 1455 | 20272 | 282359 | ... | U(-1,0,6)14. |

__The second question__

is about triangular division and thus about equations of the type a

^{2}+a = k(b

^{2}+b).

There are no solutions if k is a square. For non-square 'k', solutions approach the square root of 'k'.

Here are some examples:

__a__

^{2}+a = 2(b^{2}+b)a: | 0 | 3 | 20 | 119 | 696 | 4059 | 23660 | ... | U(0,3,2)6. |

b: | 0 | 2 | 14 | 84 | 492 | 2870 | 16730 | ... | U(0,2,2)6. |

__a__

^{2}+a = 3(b^{2}+b)a: | 0 | 2 | 9 | 35 | 132 | 494 | 1845 | ... | U(0,2,1)4. |

b: | 0 | 1 | 5 | 20 | 76 | 285 | 1065 | ... | U(0,1,1)4. |

__a__

^{2}+a = 5(b^{2}+b)a: | 0 | 5 | 14 | 99 | 260 | 1785 | 4674 | 32039 | 83880 | 574925 | 1505174 | ... | U(0,14,8)18 combined with U(-1,5,8)18. |

b: | 0 | 2 | 6 | 44 | 116 | 798 | 2090 | 14328 | 37512 | 257114 | 673134 | ... | U(0,6,8)18 combined with U(0,2,8)18. |

__a__

^{2}+a = 6(b^{2}+b)a: | 0 | 3 | 8 | 35 | 84 | 351 | 836 | 3479 | 8280 | 34443 | 81968 | 340955 | ... | U(0,8,4)10 combined with U(-1,3,4)10. |

b: | 0 | 1 | 3 | 14 | 34 | 143 | 341 | 1420 | 3380 | 14061 | 33463 | 139194 | ... | U(0,3,4)10 combined with U(0,1,4)10. |