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In mathematics, arecurrence relationis an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

My technical question is whether "each further term" is a function of only the values of the preceeding terms, or is it also a function of the index of the "further term"?

For example, if we are given that the values of two consecutive terms are (in order) ##x## and ##y## and given the equation ## z = x + 2y## where ##z## is the next consecutive term, then we don't have to know that ##z## is the 5th term or the 24th term etc. in order to find ##z## from ##x## and ##y##.

However, if we are given an equation like ##z_n = 2 z_{n-1} - 3 n z_{n-2} ## then I can find the "further term" ##z_n## if am given the value of the the previous two terms and the index ##n##, but it isn't obvious that I can find an an arbitrary term ##z## from previous terms ##x## and ##y## without knowing ##n##.

On the one hand, it seems natural to allow the definition of "reccurence relation" to assume that ##n## is known, because the familiar examples of recurrence relation say things like ##z_n = ...## , explicitly using ##n## like the argument of a function.

On the other hand, given any sequence of positive numbers ##a_0, a_1, a_2, ...## we would have the recurrence relation ##a_{n} = f(n) a_{n-1} ## where ## f(n) = \frac{a_n}{a_{n-1}} ##. So sequences of numbers satisfying recurrence relations would not be particularly special.

It might be possible to leave dependence on ##n## out of the definition of recurrence relation and still have equations like ##z_n = 2z_{n-1} -3 nz_{n-2} ## be equivalent to recurrence relations if we can deduce the value of ##n## involved from the values of the two consecutive terms. The recurrence relation would be an equation employing a function that was implemented as a complicated algorithm - to the effect of "given x and y and the first two terms ##z_0, z_1## of the sequence, perform the following steps to deduce the index ##n## ...and then set ##z =2x + 3ny##".