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This paper deals with the solutions, stability character, and asymptotic behavior of the difference equationx    n  +  1    =    α    β  +  γ    x    n  −  k;    n  ∈    N    0    ,whereN    0    =  N  ∪    0    ,  α  ,  β  ,  γ  ,  ∈    R    ∗    +and the initial values  x  − k  , x  − k  + 1 ,…, x  0  are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences