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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 ...
The n th digit of the word is + ⌊ ⌋ ⌊ (+) ⌋ where is the golden ratio and ⌊ ⌋ is the floor function (sequence A003849 in the OEIS). As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope 1 / φ {\displaystyle 1/\varphi } or φ − 1 {\displaystyle \varphi -1} .
The maximum period of lagged Fibonacci generators depends on the binary operation .If addition or subtraction is used, the maximum period is (2 k − 1) × 2 M−1.If multiplication is used, the maximum period is (2 k − 1) × 2 M−3, or 1/4 of period of the additive case.
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.
The golden spiral is a logarithmic spiral that grows outward by a factor of the golden ratio for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to Fibonacci numbers.
Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number + ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ , golden mean base , phi-base , or, colloquially, phinary .
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A random Fibonacci sequence is an integer random sequence given by the numbers for natural numbers, where = = and the subsequent terms are chosen randomly according to the random recurrence relation = {+,;,. An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence ...