I was thinking last night about the Fibonacci Numbers, as one does, and came up with these recurrence relations:

f(n) = | 0 | ... n = 0 | |

1 | ... n = 1 | ||

f(n-1) + f(n-2) | ... n > 1 | ||

f(n+2) - f(n+1) | ... n < 0 | ||

Ϝ(n) = | |f(n)| = f(|n|) | ||

S(n) = | f(n+2) - 1 = f(n) + f(n+1) - 1 | ||

Σ(n) = | ∑_{{i=0..n}} f(i) = | S(n) | ... n ≥ 0 |

0 - S(n-1) | ... n < 0 |

It turns out that the first sequence, f(n), is the good old Fibonacci Sequence, generalised to negative numbers. The others are mildly absurd, but that is appropriate for one who is the Prophet of a Religion of Discord.

Good Lord Omar said: *all things happen in fives, or are divisible by or are multiples of five, or are somehow directly or indirectly appropriate to 5.*

- The fifth Fibonacci number:
*f(5)*is 5. Coincidence? Probably. - Then, the sum of the first five Fibonacci numbers:
*S(5)*is 12. As it would happen, the 12^{th}Fibonacci number:*f(12)*has the value of 12×12. Coincidence? Yes! - Then, the sum of the first 12 Fibonacci numbers minus 12:
*S(12) - 12*gives 364. As it would happen, this is the precisely number of days in a year, according to the Religion of the Prophet St. Matty. Coincidence? Absolutely!

But according to the Prophet, all coincidences are the will of the Goddess. And that is as it should be.

If you're interested in some values of the relations, and are too lazy to work them out yourself, here are a few divinely generated by the holy prophet himself:

n | ... | -12 | -11 | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ... |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

f(n) | ... | -144 | 89 | -55 | 34 | -21 | 13 | -8 | 5 | -3 | 2 | -1 | 1 | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | ... |

Ϝ(n) | ... | 144 | 89 | 55 | 34 | 21 | 13 | 8 | 5 | 3 | 2 | 1 | 1 | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | ... |

S(n) | ... | -56 | 33 | -22 | 12 | -9 | 4 | -4 | 1 | -2 | 0 | -1 | 0 | 0 | 1 | 2 | 4 | 7 | 12 | 20 | 33 | 54 | 88 | 143 | 232 | 376 | ... |

Σ(n) | ... | -88 | 56 | -33 | 22 | -12 | 9 | -4 | 4 | -1 | 2 | 0 | 1 | 0 | 1 | 2 | 4 | 7 | 12 | 20 | 33 | 54 | 88 | 143 | 232 | 376 | ... |

Any mistakes are intentional, and must be accepted as dogmatic fact.

I love creating my own religion.

... Matty /<