Let and , where . Then a theorem of Carlitz et al. states that each function f, composed of several a's and b's, can be expressed in the form c1a + c2b - c3, where c1 and c2 are consecutive Fibonacci numbers determined by the numbers of a's and of b's composing f and c3 is a nonnegative constant. We provide generalizations of this theorem to two infinite families of complementary pairs of Beatty sequences. The particular case involving `Narayana' numbers is examined in depth. The details reveal that , with n nested pairs of , is a 7th-order linear recurrence, where is the dominant zero of x3 - x2 - 1.
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