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the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377. The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci [17] [18] where it is used to calculate the growth of ...
One can consider Fibonacci integer sequences and take them modulo n, or put differently, consider Fibonacci sequences in the ring Z/nZ. The period is a divisor of π( n ). The number of occurrences of 0 per cycle is 0, 1, 2, or 4.
The semi-Fibonacci sequence (sequence A030067 in the OEIS) is defined via the same recursion for odd-indexed terms (+) = + and () =, but for even indices () = (), . The bisection A030068 of odd-indexed terms s ( n ) = a ( 2 n − 1 ) {\displaystyle s(n)=a(2n-1)} therefore verifies s ( n + 1 ) = s ( n ) + a ( n ) {\displaystyle s(n+1)=s(n)+a(n ...
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, ... .
A Pell–Wieferich prime is a prime p satisfying p 2 divides P p−1, when p congruent to 1 or 7 (mod 8), or p 2 divides P p+1, when p congruent to 3 or 5 (mod 8), where P n denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 10 9 (sequence A238736 in the OEIS). In fact, Pell ...
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.
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 ...
The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two. Hasse diagram of some subclasses of constant-recursive sequences, ordered by inclusion In mathematics , an infinite sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is called constant ...
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