y1 = 1, y2 = 3, y3 = 5. That second example illustrates that the first n odd numbers add to n^2.
I don't know how to derive n^2 by this matrix.
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https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain/733805?r=SearchResults#733805 – K.defaoite Oct 09 '20 at 15:52
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1You could work backwards: the inverse of $S$ is clearly the matrix with $1$ on the maindiagonal, and $-1$ on the "diagonal" below. Then is you apply this $S^{-1}$ to the vector $1,4,\dots, k^2, \dots, n^2$ you'll get in the $k$-place $(k+1)^2-k^2=2k+1$. – ancient mathematician Oct 09 '20 at 16:02
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Just work with $n\times n$ matrix. – cosmo5 Oct 09 '20 at 17:47