I have a series of the form $S_1=x(x+1)$ and another series $S_1/k$, for any $k \in \mathbb{N}$. Now I want to find the values where the elements of two series are equal. For example, let $k$ be 3, then the intersection of the two series gives $2,30,420,5852,81510,1135290$.

In other words, what is the condition for which a scaled down version of a triangular number is also a triangular number? Can a formula be derived to find these ways without resorting to equating the two series and deriving quadratic roots i.e. using series formulae or triangular numbers etc.?

  • $\begingroup$ Why you need a general term? It's not always possible to have a simple general $n$th term in terms of $n$. $\endgroup$ – tarit goswami Apr 18 at 14:18

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