<s>[INST] interpretation on $1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}$ and $1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}$ How to interpret the results
$$
1^2+2^2+\ldots+n^2=\binom{n+1}{2}+2\binom{n+1}{3}
\\
1^3+2^3+\ldots+n^3=\binom{n+1}{2}+6\binom{n+1}{3}+6\binom{n+1}{4}
$$
I want to find a clear argument (combinatorial example,etc.) to prove this, other than induction or merely use the formula.
 [/INST] Two different combinatorial interpretations of your first identity are given in the answers to this question. My answer (not the accepted one) was based on counting the number of ways you can place two white bishops on an $n\times n$ chessboard so that they guard each other. Calling this number $B_n$, and counting in two different ways, I showed that
$$B_n=2\binom{n+1}3+2\binom n3$$
and
$$B_n=2[1^2+2^2+\cdots+(n-1)^2].$$
For details see my answer to that old question.
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