( [nφ]_(n>1) and ( [nφ~2] )_(n>1) are well-known complementary Beatty sequences. An infinite set of complementary Beatty sequences, based on a generalization of ratios of Fibonacci numbers and higher powers of φ, is proved. An open problem posed by Clark Kimberling, the Swappage Problem, is resolved in the affirmative as a special case of this set of complementary Beatty sequences.
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