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Sequences Obeying Certain Conditions Modulo 4

Source: Irish Mathematical Olympiad - Paper 2 - 2008 -Question 4

May 11, 2008
linear algebramatrixnumber theory proposednumber theory

Problem Statement

Given k∈[0,1,2,3] k \in [0,1,2,3] and a positive integer n n, let fk(n) f_k(n) be the number of sequences x1,...,xn, x_1,...,x_n, where x_i \in [\minus{}1,0,1] for i\equal{}1,...,n, and x_1\plus{}...\plus{}x_n \equiv k mod 4 a) Prove that f_1(n) \equal{} f_3(n) for all positive integers n n. (b) Prove that f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4 for all positive integers n n.
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