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2. Pell's equation - Wikipedia

   en.wikipedia.org/wiki/Pell's_equation

   Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √ 2. The numbers x and y appearing in these approximations, called side and diameter numbers, were known to the Pythagoreans, and Proclus observed that in the opposite direction these numbers obeyed one of these two equations. [5]

3. Pell number - Wikipedia

   en.wikipedia.org/wiki/Pell_number

   In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and 82 = 2 × 34 + 14 = 70 + 12.

4. Collatz conjecture - Wikipedia

   en.wikipedia.org/wiki/Collatz_conjecture

   If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average ⁠ 3 / 4 ⁠ of the previous one. [16] (More precisely, the geometric mean of the ratios of outcomes is ⁠ 3 / 4 ⁠.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run ...

5. Farey sequence - Wikipedia

   en.wikipedia.org/wiki/Farey_sequence

   The latter means that, for Farey sequences of even order n, the number of fractions with numerators equal to ⁠ n / 2 ⁠ is the same as the number of fractions with denominators equal to ⁠ n / 2 ⁠, that is (/) = (/).

6. List of integer sequences - Wikipedia

   en.wikipedia.org/wiki/List_of_integer_sequences

   "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.

7. Continued fraction - Wikipedia

   en.wikipedia.org/wiki/Continued_fraction

   The two sequences {Τ 2n−1} and {Τ 2n} might themselves define two convergent continued fractions that have two different values, x odd and x even. In this case the continued fraction defined by the sequence { Τ n } diverges by oscillation between two distinct limit points.