Problem of the Week

PROBLEM 226

A sequence x₁, x₂, x₃,... is called Fibonacci-like if it satisfies the property: x_k+ x_k+1 = x_k+2, for k ≥ 1.  The classic Fibonacci sequence is Fibonacci-like with initial condition x₁= 1, x₂ = 1, leading to the sequence 1,1,2,3,5,8,13,21,34,...; the Lucas numbers are Fibonacci-like with initial condition x₁= 1, x₂ = 3, leading to the sequence 1,3,4,7,11,18,29,47,76,... .

Which Fibonacci-like sequence is the longest before seeing the term 2005?

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ã2005 Alberto L. Delgado