Having 5+5+5 possible combinations doesn't mean that a "nice sequence" is a sequence of all these 125 colors.

1) 444, 443, 434, 344, are different colors, but the change is quite small for our eyes to see the absolute color difference. So a user stepping through colors to select one may decide that 125 is too many alternatives. But at the same time, the relative change is large enough that you can't get soft transitions. There is a huge difference between our eyes capacity for absolute color and relative color.

2) A "nice sequence" may potentially contain the same color multiple times. while the sequence steps around between different colors. But walking between colors does not require a lookup table to store all steps. It's enough with a table that contains the start and end colors of a transition and what curve/color space to use for the intermediate steps. And a "nice sequence" should normally not be too repetitive, which means that it shouldn't continuously do the exact same thing. So it's "nicer" to decide on maybe 8-16 target colors and then have a display randomly select one of the target colors and do a transition from current color to target color and then select the next target color randomly. In a color wheel, you get a "ball" that randomly visits different positions on the rim of the color wheel.

If you really do want the user to see the same sequence all the time then it's easier to just walk the wheel rim. Just change the angle in a HSV color space and continuously do the color transitions that you can see in Andy's linked image. A bit similar to walking the rainbow colors with the difference that the rim of the color wheel forms a loop so your first and last color is the same.

3) If the goal is for a user to select a color by using buttons or commands, then it's best to separate the color selection into the HSV parameters. Let the user select the hue separately from saturation and value. Or maybe it isn't even interesting to let the user select the intensity so maybe the user should only get a choice to select the position on the two-dimensional color wheel instead of also selecting the "height" i.e. the distance from the apex on the HSV cone.

But whenever you make a post, you always end up missing to answer the "why" part, just resulting in more and more open questions.

You still haven't told us why you think it's important to have a 125-entry table spanning "all" your colors.

And you haven't told us your new concept - what is a "nice sequence"?

And you have not asked us what problem you are trying to solve. An end user selecting a color? A device randomly selecting a fixed color for a number of seconds/minutes before selecting a different color? A device constantly gliding between colors (which do require way more than 125 colors to not look very irritatingly uggly)? Something else? A device that manually or automatically must be able to care about surrounding light levels and adjust the power output to the LEDs depending on if it is dark or light in the room? Anything else?

Don't you know what you are trying to solve? Or is the goal just to keep the forum members busy trying to second-guess you?