How to make my iterated function calculate more quickly?

Sn[n_] := x /; 0 <= n <= 8
Sn[n_] := 0 /; n > 8

Tn[n_] := 1 /; 0 <= n <= 8
Tn[n_] := 0 /; n > 8

Un[n_] := x /; 0 <= n <= 8
Un[n_] := 0 /; n > 8


Z[n_] := n^2 + n;

an[0] = 1;

an[n_] := 
  an[n] = -1/
    Z[n] Sum[((k)*(k - 1)*Sn[n - k] + (k)*Tn[n - k] + Un[n - k])*
      an[k], {k, 0, n - 1, 1}];
an[500]

Some friends say my last code was not clear,I am very sorry.Here I rewrite the code. This code calcute too slowly,please help me!Thank you!

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functions - Get leading series expansion term?

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plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

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