Does a sum of squares become smaller as the number of terms increases?

Question

I am interested in the following question:

Let $,kn$ be a positive integeres.

Assume $\sum_{i=1}^{k} L_i=\sum_{i=1}^{k+1} \tilde L_i=n$, where $L_i,\tilde L_i$ are positive integers.

Is it true that $\sum_{i=1}^{k} L_i^2 > \sum_{i=1}^{k+1} (\tilde L_i)^2$?

(i.e does the sum of squares necessarily get smaller when the number of terms increase?)

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The motivation actually comes from this question of mine. In an answer there, it is proved that the dimension of a certain subspace of matrices depends on a sum of squares of this form.

Answer 1

No.

$$1^2+1^2+6^2=38$$

but

$$4^2+4^2=32.$$