17. Let bo 1, b2 = 1, and b-4. Use generating functions to solve the recurrence equation bn+3-4bn...
15. Use generating functions to solve the recurrence equation rn = rn-1 + 6rn-2 for n > 2 with ro = 1 and rı = 3.
8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0 (a) Find the generating function for an and then solve for an b) What is the homogeneous recurrence relation that an satisfies? (c) Repeat (a) and (b) for bn 72.
8. Consider the following simultaneous homogeneous recurrence relations: 3a-12bn-1 bn-an-1 + 2bn-1 for n > 1, with initial conditions ao 1 and bo - 0...
1) Use Generating Functions to solve each of the following recurrence relations: (a) a(n)=2a(n-1)-a(n-2) if n>1, while a(0)=2, a(1)=1
Use generating functions to solve the following recurrence. T(0) = 0, T(1) = 1. T(n) = 7 T(n-1) – 12 T(n-2)
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
(8 marks) Suppose that bo, bi,b2,... is a sequence defined as follows: bo 1, b 2, b2 3, and b bk-1 + 4bk-2 +5bk-3 for all integers k 2 3. Prove by mathematical induction that bn S 3" for all integers n 2 0.
solve part a and b it is from from Topics in
Combinatorics
Problem 4. Recall that F(x) 20+1x+1x2+2x3+3x4 +... 1-x -x2 is the ordinary generating function for the Fibonacci sequence Part A. Find explicit ordinary generating functions for 2m, n 2m n(0, n 2m+ 1 1 And HnF2n Part B. Finding a generating function for Kn F2 is trickier. Start with the equations Add these to develop a recurrence for K, and use that recurrence along with appropriate initial conditions,...
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion defined by bo - 1 and bn- k-0 n E N. Show that bn-- Hint: Use a) with e*e*1 and the inverse of a power series found in the lecture.
Let e-Σ (Application of Cauchy product) for x e R. Exercise 21: n-0 a) Show that bk for all b) Let (bn)neNo be the recursion...
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= = Exercise 2.4. Define the functions f(Bo, B1, B2) 21–1(Y; – Bo – B1Xi – B2X?)2 and g(B0, B1) 2_1(Y; – Bo – B1X;)?, and let (Bo, ß1, B2) be the minimiser of f($0, B1, B2) and (Bo, Bi) the minimiser of g(Bo, B1). Explain, or prove, that 05 f(Bo, ß1, B2) <g(Bo, Bi).
Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the exponential generating functions for the follow- ing sequences in terms of f(x): (a) fan cl (b) foan (c (nani (e) 0, a,a, , (g) ao,0, a2,0, a,0,... (h) a, a2, a,... 8. (a) A sequence a satisfies the recurrence relation a3an+2, ao0 Find the exponential generating function ΣΧ0Lnz"
Section 1.7: 4. Let f(x) be the exponential generating funcion of a sequence {%). Find the...