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Research Report SRR97-003

On sequences of events with repetitions

J. Gani

Abstract: This note examines the number $t_n(k)$ of sequences of $n=1,2,3,\ldots$ , independent trials each of which may result in one of $k \geq 2$ mutually exclusive events $A_1,A_2,\ldots,A_k$ , but not containing a repetition $A_i A_i$ of the particular event $A_i$ . It is pointed out that for k=2 it is known that $t_n(2)$ is the Fibonacci number $F_{n+2}$ . We obtain an explicit result for $t_n(k)$ , and then generalize this to the number of sequences $t_{m,n}(k)$ not containing an m-repetition $A_i A_i \ldots A_i$ $\,(m \geq 2)$ of the event $A_i$ .

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