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I have been reading one of Lagrange's works where he mentions palindromic sequences in the context of Periodic continued fractions. I am wondering if that was the first time that palindromic sequences were defined?

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According to Fowler's Ratio in Early Greek Mathematics, palindromic patterns in continued fractions of $\sqrt{p}:\sqrt{q}$ with primes $p>q$ were likely known already to Pythagoreans. In the modern times the interest in continued fractions was revived largely due to Euler's work (from 1731), although Wallis and Huygens worked on them earlier, see Christiaan Huygens’ Planetarium. The palindromic pattern is noted in Euler's De usu novi algorithmi in problemate Pelliano solvendo (presented 1759). Lagrange followed Euler's lead starting in 1768, see Brezinski, History of Continued Fractions and Padé Approximants, p. 108.

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