Consecutive terms of any coprime-initial Fibonacci-like sequence have an explicit closed-form solution to ax+by=(a-1)(b-1)/2 or its +1 variant, given by six formulas depending on n mod 6.
Beiter, The midterm coefficient of the cyclotomic polynomialΦ pq(X),Amer
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A Pair of Diophantine Equations and Fibonacci-Like Sequences
Consecutive terms of any coprime-initial Fibonacci-like sequence have an explicit closed-form solution to ax+by=(a-1)(b-1)/2 or its +1 variant, given by six formulas depending on n mod 6.