MIT6_042JS10_lec32

# MIT6_042JS10_lec32 - Mathematics for Computer Science MIT...

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1 Albert R Meyer, April 26, 2010 Mathematics for Computer Science MIT 6.042J/18.062J Generating Functions for Recurrences lec 12M.1 Albert R Meyer, April 26, 2010 The Rabbit Population ! A breeding pair of rabbits produces a newborn pair every month. ! Rabbits breed after one month. ! After n months: w n ::= # ne w born pairs b n ::= # b reeding pairs ! Start with a newborn pair: w 0 = 1 b 0 = 0 lec 12M.7 Albert R Meyer, April 26, 2010 The Rabbit Population w n ::= # ne w born pairs b n ::= # b reeding pairs lec 12M.8 Therefore, Albert R Meyer, April 26, 2010 The Rabbit Population b n = b n-1 + b n-2 It was Fibonacci who was studying rabbit population growth! lec 12M.9 Albert R Meyer, April 26, 2010 Generating Function for Rabbits lec 12M.10 Albert R Meyer, April 26, 2010 Generating Function for Rabbits lec 12M.11

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2 Albert R Meyer, April 26, 2010 lec 12M.12 Generating Function for Rabbits Albert R Meyer, April 26, 2010 [ x n ] 1 (1 ! ± x) = ± n Coefficient notation lec 12M.13 [ x n ] B(x) :: = b n = coeff of x n in power series for B(x) Albert R Meyer, April 26, 2010 Generating Function for Rabbits use
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MIT6_042JS10_lec32 - Mathematics for Computer Science MIT...

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