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Let $W=(e_0-e_{15}, e_0+e_1, e_1+e_2, \dots , e_{14}+e_{15})$ be a 16-dimensional vector in $\mathbb{Z}^{16}$ with each $e_i \in \{-k, \dots , k\}$ for some positive integer $k$.

Given that the $e_i$s are i.i.d., we wish to find the distribution of $\|W\|^2$; namely $$ (e_0-e_{15})^2+(e_0+e_1)^2+ \dots + (e_{14}+e_{15})^2 $$

Doing this using computer simulations is a bit difficult because $ k $ can have a value of $ 8 $ and it gets so heavy on the machine.

Any suggestions?

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  • $\begingroup$ What's the story with 8 being heavy on the machine? It can slip and becomy infinity? $\endgroup$
    – zhoraster
    Nov 23, 2021 at 10:06
  • $\begingroup$ @zhoraster Since $k=8$, then I have to do a FOR loop with $17^{16} \approx 2^{64}$ calculations. $\endgroup$
    – C.S.
    Nov 24, 2021 at 11:13

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