Only with the third do we have two pairs (two adults and two babies), 3 pairs the fourth month (the first pair of adults with another pair of babies, but the second generation still without issue). This is because he builds in a highly realistic maturation delay right from the start: 1 pair the first month, and still just one pair the second month. At 12 months this would lead to 2048 pairs of rabbits, compared to Fibonacci's much more modest 144. What Fibonacci does refrain from assuming is a geometric population expansion such as 1 pair the first month, 2 the second, 4 the third, and so on. And the birth of two babies of the same sex to one adult pair could be offset by two of the opposite sex being born to another. True, he ignores potential in-breeding problems and mortality, but this isn't unduly optimistic given that he asked what quantity of rabbits could be expected at the end of only 12 months. But is it fair of Rachel to describe them as a 'highly unbelievable breeding process'? Yes, Fibonacci made certain assumptions, as we all do when figuring out what to expect. See 'Fibonacci's fast and slow breeders'. I still hope my other results in this thread are interesting, especially the quantitative and structural differences in sequences resulting from varying the maturation delay. That's effectively all I did in multiplying each item in the standard Fibonacci 1 1 2 3 5 8 by 8 for example to get 8 8 16 24 40 64 without spotting its triviality at the time.Īnyway I'm very pleased you had a look. For example 1 23 7 13 4 5 each multiplied by 4 is going to result in 4 92 28 52 16 20. Obviously they're going to generate a new sequence which includes a number which is a perfect square of the first. Tell someone it's a 'sequence' and that they are to multiply each successive number (including the 1) by any one number already there in the sequence. Beginning with 1, line them up in otherwise any way you like. Take any random collection of numbers that includes 1. I liked it too at first, Rachel, but now I realise it's not so exciting after all, since it says nothing about Fibonacci numbers in particular unfortunately. Lastly although I came across these results concerning Fibonacci powers on my own (see also my previous comment about 5), I daresay they aren't new discoveries. For example 13 takes 7 numbers to get to its square, and 13 occupies position 7. Note also the number of numbers in each sequence, which is equal to the position of the start number in the standard Fibonacci sequence. This can be remedied perhaps with: 0 1 1) (My treatment of 1 seems a bit anomalous here since although it 's a perfect square, I haven't presented it as the result of any addition. In fact the only start numbers we can hit the square with seem to be the Fibonaccis and no others: But then 4 and 6 aren't in the Fibonacci sequence either. The same goes for 6: 6 6 12 18 30 48 doesn't include 36. Try this with 4 however: 4 4 8 12 20, and we don't land on 4 squared. Now let's pretend 5 is the first Fibonacci number instead of the usual 1, but still use the same addition algorithm: 5 5 10 15 25. To find out more read The life and numbers of Fibonacci. The sequence is also closely related to a famous number called the golden ratio. You can find it, for example, in the turns of natural spirals, in plants, and in the family tree of bees. Real rabbits don't breed as Fibonacci hypothesised, but his sequence still appears frequently in nature, as it seems to capture some aspect of growth. And from that we can see that after twelve months there will be pairs of rabbits.

Starting with one pair, the sequence we generate is exactly the sequence at the start of this article.

Therefore, the total number of pairs of rabbits (adult+baby) in a particular month is the sum of the total pairs of rabbits in the previous two months: Writing for the number of baby pairs in the month, this gives Writing for the number of adult pairs in the month and for the total number of pairs in the month, this givesįibonacci also realised that the number of baby pairs in a given month is the number of adult pairs in the previous month. He realised that the number of adult pairs in a given month is the total number of rabbits (both adults and babies) in the previous month. Fibonacci asked how many rabbits a single pair can produce after a year with this highly unbelievable breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month).

Natural fibonacci sequence