Color Necklace Count Calculator

For every cyclic shift, count fixed colorings as k raised to gcd(n,shift), sum them, and divide by the number of rotations n. The indexing and counted objects are stated in the method and worked example. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

Reflections remain distinct; identifying flips as well would count bracelets and requires a different group action. Small boundary cases and the displayed identity provide useful independent checks. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

Length is capped at 40 and colors at 8; unsafe orbit counts are refused. The implementation uses integer arithmetic internally and refuses results beyond the safe display boundary. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.