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Let  F n   be the  n th Fibonacci number. For 1 ≤  k  ≤  m , let[mk]    F   =F   mF    m  -  1    ⋯    F    m  -  k  +  1F   1   ⋯    F   k$$\left[ {\matrix{m \hfill \cr k \hfill \cr } } \right]_F = {{F_m F_{m - 1} \cdots F_{m - k + 1} } \over {F_1 \cdots F_k }}$$be the corresponding Fibonomial coefficient. It is known that[mk]    F    $\left[ {\matrix{m \hfill \cr k \hfill \cr } } \right]_F $$$    is a Fibonacci number if and only if either  k  = 1 or  m  ∈ { k ,  k  + 1}. In this note, we find all solutions of the Diophantine equation[mk]    F   ±  1  =    F   n    $\left[ {\matrix{m \hfill \cr k \hfill \cr } } \right]_F \pm 1 = F_n $   .

Fibonomial coefficients at most one away from Fibonacci numbers