Recurrence Relation Models

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Recurrence Relation Models
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• A recurrence relation is a recursive formula that
  counts the number of ways to do a procedure
  involving n objects in terms of the number of
  ways to do it with fewer objects.
• E.g., an c1an-1 c2an-2 , a1 0, a2 1
• A recurrence relations starting values are
  called initial conditions.

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• Proving things about a recurrence relation
  usually is done by mathematical induction.
• Typical forms of recurrence relations include
• an c1an-1 c2an-2 . . . cran-r
• an c1an-1 c2
• an c1an-1 f(n)

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• Solve a recurrence relation for a particular
  value of n by
• Computing the values by hand/calculator, starting
  from initial conditions.
• Writing an iterative program to compute the
  values, starting from initial conditions.
• Deriving a formula for the recurrence, and
  computing the value directly with it.

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Example 1 Permutations

• Let an be the number of ways to permute n
  objects.
• Give a recurrence relation for an.
• What is a1?

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Example 2 The Fibonacci Relation

• Initially, there is 1 pair (i.e., male female)
  of newborn rabbits.
• Every month, each pair of rabbits that are over 1
  month old produce a new pair.
• How many rabbits are there after 12 months?

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• Let Fn the of rabbits after n months.
• Let F0 1, the pair is just born.
• F1 1, the pair is 1 month old.
• F2 2, the 1st pair produces a 2nd pair.
• F3 3, the 1st pair produces a 3rd pair.
• F4 5, the 1st 2nd pair each produce a pair.
• Fn Fn-1 Fn-2 , all pairs born 2 months ago
  produce new pairs.

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Compounding Interest

• Let P be the principal borrowed from a bank.
• Let r be the interest rate per period.
• Let an represent the amount due after n periods.
• What is a0 ?
• What is a recurrence for an ?
• How would it change, if D dollars were paid to
  the bank at the
• End of every month?
• Beginning of every month?

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Distributing money!

• Find a recurrence relation for an the number of
  ways to distribute either a 1 bill, a 5 bill, a
  10 bill, or a 20 dollar bill on successive days
  until a total of n dollars has been distributed.
• What is a a1 ?
• What is a a2 ?
• What is a a5 ?
• What is a an ?

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• Suppose Bill Ashby gives 1 prize away each day,
  where there are
• 3 kinds of prizes worth 5
• 7 kinds of prizes worth 10.
• Give a recurrence pn for the number of ways for
  Bill to give away n worth of prizes.
• What is p1? p5? p10? pn?

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• If n non-parallel lines are drawn in the plane,
  no 3 lines intersecting at the same point, how
  many regions rn do these lines divide the plane?
• What is r0? r1? r2? r3? rn?

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Tower of Hanoi
Ending position
Starting position
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Tower of Hanoi ...

• Move the discs from the left peg to the center
  peg, preserving their size order when you are
  done such that
• Move the top disc from 1 peg place it on the top
  of a pile of another peg.
• At no time may you move a larger disc on top of a
  smaller disc (herniating the smaller disc).

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Tower of Hanoi ...

• Let mn denote the number of moves needed in an
  n-disk game.
• What is m1?
• What is m2?
• What is m3?
• What is mn?
• Prove that mn 2n - 1.

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Lancaster Equations of Combat

• 2 Armies A B engage in combat.
• Let a0 b0 denote the of soldiers alive before
  combat begins.
• Let an and bn be the of soldiers alive after
  the nth day of combat.
• The daily decrease in each army equals 1/2 the
  size of the opposing army (or 0, whichever is
  larger).
• What is a1 b1? a2 b2? an bn?

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Forbidden subsequences

• Find a recurrence relation for an, the of
  n-digit ternary sequences without the subsequence
  012.
• What is a1? a2? a3? a4? a5? a6? an?
• Partition all such sequences into 3 parts
• Those that begin with 1
• Those that begin with 2
• Those that begin with 0
• For this case, subtract the of otherwise good
  sequences that begin with 012. How many are
  there?

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Characters

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