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We have a sequence where the power of two (say) decreases and the power of three (say) increases. $$2^{r-i} 3^i, \text{for } 0\leq i \leq r$$ Is there a name for this?

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You can write it as $$2^r\Bigl(\frac32\Bigr)^i\ ,$$ which shows that it is a geometric sequence with first term $2^r$ and ratio $\frac32$.