Using an integer relations algorithm, we get,


$$6F_{4n}= -F_{n-2}^4-3F_{n-1}^4+3F_{n+1}^4+F_{n+2}^4$$

The pattern of the subscripts is clear. Expressing the coefficients as a number triangle and including higher powers,

$$F_{2n} = -1,\,1$$

$$6F_{4n} = -1,\,\color{brown}{-3},\,3,\,1$$

$$120F_{6n} = -1,\,\color{brown}{-4},\,20,\,-20,\,4,\,1$$

$$21840F_{8n} = -1,\,\color{brown}{-14},\,91,\,364,\,-364,\,-91,\,14,\,1$$

$$24504480F_{10n} = -1,\,\color{brown}{-33},\, 748,\, 3927,\, -17017,\,17017, \dots\quad\quad\quad\quad$$

Question: Anybody knows the formula for the coefficients?

P.S. One pattern easy to spot is $F_n\,L_{n+1}=0,3,4,14,33,\dots$

See this related post for the slightly different form for odd powers (it contains both the $F_n$ and $F_{pn}$ terms), and Ron Knott's article on fibonomials.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.