I have a question; help is welcome.(adsbygoogle = window.adsbygoogle || []).push({});

Let s_{n}be a linear, non-homogeneous recurrence sequence, and let h_{n}be a corresponding homogeneous sequence (with initial values to be determined).

As it turns out, the offset between the two (s_{n}- h_{n}) is given by the steady state value of s_{n}, if the initial values of h_{n}are offset from those of s_{n}by the same amount. Precisely what is the reason for that?

This steady state value bears no relation to the initial values of the sequence s_{n}; more properly, it should be called the steady state value of the recurrence rule. I believe it is clear that a linear recurrence rule will have exactly one steady state value, neither none nor multiple (as the steady state is given by the root of a first-degree polynomial). And the steady state of h_{n}is, of course, 0 (the root of a first-degree polynomial through the origin -- d'oh!). Therefore,if the desired offset does not depend on the initial values of s(the offset was hand-set on the initial values, but who says it will stay that way further on in the sequences?), then the difference of the two steady states (thus the steady state of s_{n}_{n}, as the one of h_{n}is zero) should do. But why is the partin boldtrue?

Thanks--

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# Offset between non-homogeneous and homogeneous recurrence sequences

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