For any k ≥ 2 the probability that no k+1 of n independent uniform [0,1] lengths form a (k+1)-gon is a product whose factors involve a k-step Fibonacci-type recurrence.
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Pick-up Sticks and the General Fibonacci Numbers
For any k ≥ 2 the probability that no k+1 of n independent uniform [0,1] lengths form a (k+1)-gon is a product whose factors involve a k-step Fibonacci-type recurrence.