On 16/04/2013 5:38 AM, Virgil wrote: > In article > <email@example.com>, > WM <firstname.lastname@example.org> wrote: > >> Matheology § 250 >> >> Fibonacci-sequences with fatalities >> >> The Fibonacci-sequence >> f(n) = f(n-1) + f(n-2) for n > 2 mit f(1) = f(2) = 1 >> the first recursively defined sequence in human history (Leonardo of >> Pisa, 1170 - 1240) should be well known. A pair of rabbits that >> reproduces itself monthly as from the completed second month on will >> yield 144 pairs after the first 12 months: >> 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. >> >> If we assume that each pair reproduces itself after two months for the >> last time and dies afterwards, we get a much more trivial sequencence: >> 1, 1, 1, ... >> However the rabbits behind this numbers change. If we call them in the >> somewhat unimaginative but effective manner of the old Romans, we get >> Prima, Secunda, Tertia, Quarta, Quinta, Sexta, Septima, Octavia, Nona, >> Decima and so on. >> >> A more interesting question is brought up, if the parent pair dies >> immediately after the birth of its second child pair. Then the births >> in month n can be traced back to pairs who have been born in months >> n-2 and n-3. >> g(n) = g(n-2) + g(n-3). >> The number f(n) of pairs in month n is given by those born in month n, >> i.e. g(n) and those already present in month n-1, i.e., f(n-1), minus >> those who died in month n (i.e. those who were born in month n-3: >> f(n) = g(n) + f(n-1) - g(n-3) = g(n-2) + f(n-1) >> g(n-2) = f(n) - f(n-1) >> g(n-2) = g(n-4) + g(n-5) >> = f(n-2) - f(n-3) + f(n-3) - f(n-4) >> = f(n-2) - f(n-4) >> For n > 4 we have with f(1) = 1, f(2) = 1, f(3) = 2, f(4) = 2. >> f(n) = f(n-1) + f(n-2) - f(n-4) >> The number of pairs during the first 12 months is >> 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21. >> The sequence grows less than the original one, but without enemys or >> other restrictions it will grow beyond every threshold. If we wait >> aleph_0 days (or use the trick that the duration of pregnancy is >> halved in each step, facilitated by genetic evolution), we will get >> infinitely many pairs - a nameless number, alas of nameless rabbits, >> because they cannot be distinguished. The set of all Old-Roman names >> has been exhaused already, and even all of Peano's New-Roman names S0, >> SS0, SSS0, ... have been passed over to pairs which already have >> passed away. That is amazing, since none of the pairs of the original >> and much more abundant Fibonacci sequence has to miss a name. >> >> But this sequence with fatalities can also be obtained without >> fatalities (killings), namely if each pair has to pause for two months >> after each littering in order to litter again in the following month. >> Mathematically, there is no difference. Set theory, however, yields a >> completely different limit in this case. The limit set of living >> rabbits is no longer empty, but it is infinite - and every rabbit has >> a name. > > There may be no difference in the number of rabbits but there will be a > considerable difference in both the parent-child linkages of those > rabbits and the number of dead rabbits. > > Either of which is enough to distinguish the two cases as being > mathematically different, even if not WMytheologally different.
In addition to that, there is only one person here littering: WM