Does the number of alleles present depend on the Ploidy of the organism

In diploid organisms there are two alleles in a given organism.
In triploids, would there be three alleles in a given organism? And so on?
Or are there always, for the most part, three alleles in a give organism regardless of the ploidy number?

Since the ploidy number reflects the number of chromosome "sets," there must be that number of genes; therefore there must be that number of alleles. In tetraploidy, there are 4 complete chromosomes sets, so there must be 4 alleles for each gene. However, they are not all different, since most genes have only 2 alleles. A pentaploid flower will have 5 alleles for flower color.