The Fibonacci numbers ,

obey the following *recurrence relations*,

and so on. As a number triangle, the coefficients are,

See Ron Knott’s article on the *fibonomials, *so-called since the above is reminiscent of the *binomial* triangle. However, I found *another* set of recurrence relations can be given as,

etc. As a number triangle,

Compare the two triangles. Notice how, for odd powers, the *same* coefficients appear, *though moved up by one odd power*. I have no explanation for the phenomenon, other than the fact that I’ve seen several instances already of a “recycled” polynomial appearing in many contexts.

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