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Sequences - CSE 1400 Applied Discrete Mathematics Sequences...

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CSE 1400 Applied Discrete Mathematics Sequences Department of Computer Sciences College of Engineering Florida Tech Spring 2011 1 Sequences 1 1 . 1 Operations on Sequences 2 1 . 2 Useful Sequences 3 1 . 3 Growth Rates 5 1 . 4 Defined by Recurrence Equations 6 1 . 5 Defined by Functions on the Natural Numbers 8 1 . 6 Computed by Algorithms 8 1 . 7 Non-Computable Sequence 9 Abstract 1 Sequences Sequences are ordered lists of values. Ordinal numbers: first, second, third, fourth, . . . , specify the positions of these values. One famous sequence is the Fibonacci sequence. h 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . i The first, second, third, fourth, and fifth Fibonacci numbers are 0, 1, 1, 2, and 3. The values in the Fibonacci sequence can be named f k where the subscript k denotes the position of the value. In comput- ing, it is customary to start the subscript k at 0 so that the first, sec- ond, third, fourth, and fifth Fibonacci numbers are named f 0 , f 1 , f 2 , f 3 , and f 4 . Let ~ S = h s 0 , s 1 , s 2 , s 3 , s 4 , . . . i be a sequence. The subscripted names s 0 , s 1 , s 2 , s 3 , s 4 , . . . , are called terms and they refer to values in the first, second, third, fourth, fifth, etc., positions of the sequence. In computing theory, sequences
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cse 1400 applied discrete mathematics sequences 2 contain a countably infinite number of terms. In computing practice, sequences have a finite number of terms. For instance, the sequence of primes less than 30 is ~ P < 30 = h 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 i There is a a sequence with no terms at all, called the empty sequence. I’ll use σ to denote the empty sequence, so that σ = ~ . 1 . 1 Operations on Sequences There are some basic operations that can be performed on sequences. empty() maps a sequence to True or False . null ( ~ S ) = True if ~ S is empty False if ~ S contains at least one term length() maps a finite sequence to the number of terms it contains. length ( ~ S ) = n head() maps a non-empty sequence to its first element head ( ~ S ) = s 0 last() maps a non-empty, finite sequence to its last element last ( ~ S ) = s n - 1 tail() removes the first term from a sequence tail ( ~ S ) = h s 1 , s 2 , s 3 , . . . , s n - 1 i concat(,) concatenates two finite sequences to form a single sequence concat ( ~ S , ~ T ) = h s 1 , s 2 , . . . , s n - 1 , t 0 , t 1 , . . . , t m i map(,) applies a function f to each term of a sequence map ( f , ~ S ) = h f ( s 1 ) , f ( s 2 ) , . . . , f ( n - 1 ) i
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cse 1400 applied discrete mathematics sequences 3 1 . 2 Useful Sequences Here is a list of useful sequences. • The Alice sequence ~ A = h 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . . i keeps a “tally.” • The Gauss sequence ~ G = h 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . . . i computes the sum of a “tally.” • The sequence of triangular numbers ~ T = h 0, 0, 1, 3, 6, 10, 15, 21, 28, 36, . . . i computes the number of edges in a complete graph. • The doubling sequence ~ D = h 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, . . . i computes a common growth model. • The Mersenne sequence ~ M = h 0, 1, 3, 7, 15, 31, 63, 127, 255, 511 . . . i computes the sum of the doubling growth model.
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