Values
The Wythoff array has the values
- (sequence A035513 in the OEIS ).
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
The Wythoff array was first defined by Morrison (1980) using Wythoff pairs, the coordinates of winning positions in Wythoff's game. It can also be defined using Fibonacci numbers and Zeckendorf's theorem, or directly from the golden ratio and the recurrence relation defining the Fibonacci numbers.
The Wythoff array has the values
Inspired by a similar Stolarsky array previously defined by Stolarsky (1977), Morrison (1980) defined the Wythoff array as follows. Let denote the golden ratio; then the th winning position in Wythoff's game is given by the pair of positive integers , where the numbers on the left and right sides of the pair define two complementary Beatty sequences that together include each positive integer exactly once. Morrison defines the first two numbers in row of the array to be the Wythoff pair given by the equation , and where the remaining numbers in each row are determined by the Fibonacci recurrence relation. That is, if denotes the entry in row and column of the array, then
The Zeckendorf representation of any positive integer is a representation as a sum of distinct Fibonacci numbers, no two of which are consecutive in the Fibonacci sequence. As Kimberling (1995) describes, the numbers within each row of the array have Zeckendorf representation that differ by a shift operation from each other, and the numbers within each column have Zeckendorf representations that all use the same smallest Fibonacci number. In particular the entry of the array is the th smallest number whose Zeckendorf representation begins with the th Fibonacci number.
Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array ( Morrison 1980 ).
Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears in shifted form in the second row ( Morrison 1980 ).
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function, many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function, is defined as follows for nonnegative integers m and n:
In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ(n) or ϕ(n), and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.
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In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
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In mathematics, Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers.
A Fibonacci word is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
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In mathematics, the Fibonacci numbers form a sequence defined recursively by:
Willem Abraham Wythoff, born Wijthoff, was a Dutch mathematician.
In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.
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Wythoff's game is a two-player mathematical subtraction game, played with two piles of counters. Players take turns removing counters from one or both piles; when removing counters from both piles, the numbers of counters removed from each pile must be equal. The game ends when one person removes the last counter or counters, thus winning.
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In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as
