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#Life 1.05

#D Irrational 5

#D Population growth is linear with an irrational multiplier.

#D Each middleweight spaceship produced by the puffers either hits a

#D boat or is deleted by a glider.  Denoting the first possibility by

#D 1 and the second by 0, we obtain a sequence beginning 101011011010...

#D If we prepend 101, we obtain the Fibonacci string sequence, defined

#D by starting with 1 and then repeatedly replacing each 0 by 1 and each

#D 1 by 10:  1 -> 10 -> 101 -> 10110 -> 10110101 -> ...  (See Knuth's

#D "The art of computer programming, vol. 1", exercise 1.2.8.36 for

#D another definition.)  The density of 1's in this sequence is

#D (sqrt(5)-1)/2,  which implies that the population in gen t is

#D asymptotic to  (8 - 31 sqrt(5)/10) t.  More specifically, the

#D population in gen  20 F[n] - 92  (n>=6)  is 98 F[n] - 124 F[n-1] + 560,

#D where  F[n]  is the n'th Fibonacci number.  (F[0]=0,  F[1]=1,  and

#D F[n] = F[n-1] + F[n-2]  for  n>=2.)

#D Dean Hickerson, dean@ucdmath.ucdavis.edu  5/12/91

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