C L.'s periodic sequences

Probability Level 5

The sequence x0,x1,x2, x_0, x_1, x_2, \ldots is defined by the recurrence relation

xn+1=axn+bxn1+cxn2+dxn3,n3. x_{n+1} = a x_n + b x_{n-1} + c x_{n-2} + d x_{n-3}, n\ge 3.

For fixed integers a,b,c,da, b, c, d, it turns out that regardless of the initial values x0,x1,x2,x3x_0, x_1, x_2, x_3, the sequence is eventually periodic. Find the sum of all possible periods.

This problem is posed by C L.

Details and assumptions

As an explicit example, 33 is a possible period since the recurrence relation xn+1=xn2,n3x_{n+1} = x_{n-2}, n\ge 3 is eventually periodic regardless of the starting values x0,x1,x2,x3x_0, x_1, x_2, x_3.


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