Let [Formula: see text] be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that where lcm is least common multiple golden ratio. We prove for every periodic in there exists an effectively computable rational number such Moreover, we show if a independent uniformly distributed random variables then dilogarithm function.

We give multiple proofs of two formulas concerning the enumeration permutations avoiding a monotone consecutive pattern with certain value for inverse peak number or left statistic. The in both cases is given by sequence related to Fibonacci numbers. also show that there exactly one permutation whose zero among all any fixed descent composition, and we few elementary consequences this fact. Our...