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I have a recurrence relation with a complicated index pattern. However, even simple examples fail. For example, consider

RecurrenceTable[{a[n + 1] == a[n - Floor[Sqrt[n]]], a[0] == 1}, a, {n, 0, 10}]

This gives the error

"All arguments in position 1 of a[1+n]==a[n-Floor[Sqrt[n]]] should be in the form n + integer."

When I look at the type for Floor[Sqrt[n]], Mathematica tells me it is an integer, but it doesn't seem to recognize n-Floor[Sqrt[n]] as an integer. How can I use a function like Floor inside the indices for a recurrence table?

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  • $\begingroup$ Isn't the answer just a[n]=1 for all n? $\endgroup$
    – bill s
    Commented Aug 11, 2017 at 18:26
  • $\begingroup$ It might be for this one. This is just an simple example to see the error. $\endgroup$
    – Trevor
    Commented Aug 11, 2017 at 18:27

1 Answer 1

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I would not use RecurrenceTable for this... just define it recursively:

a[n_] := a[n - 1 - Floor[Sqrt[n - 1]]];
a[0] = 1;

You can verify that for this particular recursion, a[n]=1 for any n you care to chose. To see why RecurrenceTable does not work, note that the help says: "The eqns can involve objects of the form a[n+i] where i is any fixed integer." In this case, i is an integer, but it is not fixed.

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  • $\begingroup$ This is a classic example of something not working until I ask about it. I did try what you suggested based on another answer, but I must have typed something wrong somewhere. It worked this time. The question still remains about why it won't work within RecurrenceTable. $\endgroup$
    – Trevor
    Commented Aug 11, 2017 at 18:35
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    $\begingroup$ Looking at the help for RecurrenceTable, it says: "The eqns can involve objects of the form a[n+i] where i is any fixed integer." In your case, i is an integer, but it is not fixed. $\endgroup$
    – bill s
    Commented Aug 11, 2017 at 18:44
  • $\begingroup$ @bills I'd include the contents of your last comment in your answer, as it seems very relevant to the original question as it had been posed. $\endgroup$
    – MarcoB
    Commented Aug 11, 2017 at 20:26

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