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  2. Fibonacci sequence - Wikipedia

    en.wikipedia.org/wiki/Fibonacci_sequence

    Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same ...

  3. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    The ratio of Fibonacci numbers ⁠ ⁠ and ⁠ ⁠, each over ⁠ ⁠ digits, yields over ⁠ ⁠ significant digits of the golden ratio. The decimal expansion of the golden ratio ⁠ φ {\displaystyle \varphi } ⁠ [ 1 ] has been calculated to an accuracy of ten trillion ( ⁠ 1 × 10 13 = 10,000,000,000,000 {\displaystyle \textstyle 1\times ...

  4. Generalizations of Fibonacci numbers - Wikipedia

    en.wikipedia.org/wiki/Generalizations_of...

    A repfigit, or Keith number, is an integer such that, when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 (4, 7, 11, 18, 29, 47) reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the ...

  5. Fibonacci - Wikipedia

    en.wikipedia.org/wiki/Fibonacci

    In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.

  6. Lucas number - Wikipedia

    en.wikipedia.org/wiki/Lucas_number

    The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. [3] The first few Lucas numbers are 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... .

  7. Ratio - Wikipedia

    en.wikipedia.org/wiki/Ratio

    An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers: even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio.

  8. Patterns in nature - Wikipedia

    en.wikipedia.org/wiki/Patterns_in_nature

    Phyllotaxis spirals can be generated from Fibonacci ratios: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (each subsequent number being the sum of the two preceding ones). For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2.

  9. Reciprocal Fibonacci constant - Wikipedia

    en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant

    The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.