Chapter 8: Q35E (page 536)
Give a big- \(O\) estimate for the function\(f\)in Exercise\(34\)if\(f\)is an increasing function.
The required result is\(f(n) = O\left( {{n^{{{\log }_4}5}}} \right)\).
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For each of these generating functions, provide a closed formula for the sequence it determines.
a) \({\left( {{x^2} + 1} \right)^3}\)
b) \({(3x - 1)^3}\)
c) \(1/\left( {1 - 2{x^2}} \right)\)
d) \({x^2}/{(1 - x)^3}\)
e) \(x - 1 + (1/(1 - 3x))\)
f) \(\left( {1 + {x^3}} \right)/{(1 + x)^3}\)
g) \(x/\left( {1 + x + {x^2}} \right)\)
h) \({e^{3{x^2}}} - 1\)
To prove every allowable arrangement of the ndisks occurs in the solution of this variation of the puzzle.
To find number of edges and describe to make counting the edges easier.
Suppose that when is an even positive integer, and . Find
a)
b).
c).
d).
Find a closed form for the generating function for the sequence\(\left\{ {{a_n}} \right\}\), where,
a) \({a_n} = 5\) for all\(n = 0,1,2, \ldots \).
b) \({a_n} = {3^n}\)for all\(n = 0,1,2, \ldots \)
c) \({a_n} = 2\)for\(n = 3,4,5, \ldots \)and\({a_0} = {a_1} = {a_2} = 0\).
d) \({a_n} = 2n + 3\)for all\(n = 0,1,2, \ldots \)
e) \({a_n} = \left( {\begin{array}{*{20}{l}}8\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
f) \({a_n} = \left( {\begin{array}{*{20}{c}}{n + 4}\\n\end{array}} \right)\)for all\(n = 0,1,2, \ldots \)
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