Nested pure function argument as table

Question

I made a nested pure function

Nest[(x^2)/ ((2x+1) + #)&, x,n]

I would like that for each iteration the value of x increment of 1, to have something like

$$\frac{1^2}{\frac{3 + 2^2}{5 + ...}}$$

I'm wondering if this is possible?

Answer 1

You can use RecurrenceTable.

RecurrenceTable[
 {
  f[x] == (x^2)/((2 x + 1) + f[x - 1])
  , f[1] == 1/(1 + 3)
  }
 , f
 , {x, 3}]

{1/4, 16/21, 189/163}

Hope this helps.

Answer 2

I'd be tempted to implement this as a recursion:

f[x_] := f[x] = (x^2)/((2 x + 1) + f[x - 1]);
f[1] = 1/4;

Now to get the first 10 values:

f/@ Range[10]