Comments on quadratic and polynomial sequences

Suppose we have a sequence generated by a quadratic function of n.

Let's say the n-th term is A(n) = an2+bn+c, with a>0 (since otherwise the sequences would, at least eventually, have negative terms).

Consider the difference sequence generated by A(n+1)-A(n). We find A(n+1)-A(n) = (2n+1)a+b. As a result, since a is positive, we can conclude that the differences always increase at a steady rate. This means that the sound we get from this sequence is very predictable: it will have a constantly decreasing "pitch", and fairly quickly become a sequence of independent clicks once the difference grows sufficiently large.

As a result, all quadratic sequences sound essentially the same.

Here are two examples, the triangular numbers (the n-th term given by n(n+1)/2) and the square numbers (the n-th term given by n2).
OIES iddescription sound player creator
A000217triangular numbersdownloadHannah Claire Van Wyk
A000290square numbersdownloadHannah Claire Van Wyk

You can hear that they sound essentially the same. The triangular numbers are notable as the quadratic integer sequence that grows as slowly as possible: all other quadratic sequences yield sounds that drop in pitch (and become "clicky") even faster.

Further, the same is true, but more so, for sequences generated by other polynomials. They just grow too quickly to yield interesting sounds.

Many of the sequences at the OEIS are polynomial: it is probably best to explore other sequences when looking for ones with interesting sounds.