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1769 | yogev_ezra | 1 | #Life 1.05 |
2 | #D Irrational 5 |
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3 | #D Population growth is linear with an irrational multiplier. |
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4 | #D Each middleweight spaceship produced by the puffers either hits a |
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5 | #D boat or is deleted by a glider. Denoting the first possibility by |
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6 | #D 1 and the second by 0, we obtain a sequence beginning 101011011010... |
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7 | #D If we prepend 101, we obtain the Fibonacci string sequence, defined |
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8 | #D by starting with 1 and then repeatedly replacing each 0 by 1 and each |
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9 | #D 1 by 10: 1 -> 10 -> 101 -> 10110 -> 10110101 -> ... (See Knuth's |
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10 | #D "The art of computer programming, vol. 1", exercise 1.2.8.36 for |
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11 | #D another definition.) The density of 1's in this sequence is |
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12 | #D (sqrt(5)-1)/2, which implies that the population in gen t is |
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13 | #D asymptotic to (8 - 31 sqrt(5)/10) t. More specifically, the |
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14 | #D population in gen 20 F[n] - 92 (n>=6) is 98 F[n] - 124 F[n-1] + 560, |
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15 | #D where F[n] is the n'th Fibonacci number. (F[0]=0, F[1]=1, and |
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16 | #D F[n] = F[n-1] + F[n-2] for n>=2.) |
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17 | #D Dean Hickerson, dean@ucdmath.ucdavis.edu 5/12/91 |
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18 | #N |
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19 | #P -67 -32 |
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44 | #P -58 -31 |
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56 | #P -49 -16 |
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