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Araris Valerian

Periodic(ish) Polynomial(ish) Functions

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I've come across a weird class of functions, where the overall growth is polynomial bounded above and below, but there is some periodic behavior. One example is the following sequence of numbers: 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43. The 1st, 3rd, 5th, and so forth entries correspond to the function x^2+1, if x=1 corresponds to the value 2. The other entries correspond to x^2+x+1, if we match x=1 to the entry valued 3.

Has anyone seen functions/sequences like this before? I'm trying to prove that certain sequences always have this sort of periodic polynomial behavior, but I haven't seen these things before, which is making it hard to prove that an infinite set of functions falls into this category.

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Perhaps insert a term which results as 0 when x is odd/even?

Ex. y= x^2 + 1 + x*([x+1]mod 2)

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