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... as the number of ideals in a very simple poset, usually called a fence. The purpose of this note is not to prove new theorems about the sequence {Fn}. However, we wish to demonstrate that the approach has several advantages. By attaching to each Fibonacci number a geometrical object, the number gets ...
... as the number of ideals in a very simple poset, usually called a fence. The purpose of this note is not to prove new theorems about the sequence {Fn}. However, we wish to demonstrate that the approach has several advantages. By attaching to each Fibonacci number a geometrical object, the number gets ...
Recursive definitions A sequence is defined recursively if (B) some
... The examples cited above are iterative calculations to determine all the elements in the sequence up to a certain point. An alternative method to calculate a particular element, say the seventh element, would be to use the recurrence relation recursively to calculate (only the necessary) preceding v ...
... The examples cited above are iterative calculations to determine all the elements in the sequence up to a certain point. An alternative method to calculate a particular element, say the seventh element, would be to use the recurrence relation recursively to calculate (only the necessary) preceding v ...
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... is devoted to showing that this sequence is interesting. In particular, we shall look at the following properties. • n 7→ an+1 /an is a bijection between the positive natural numbers and the positive rational numbers, • n/2k 7→ an /an+2k extends to a continuous strictly increasing function on [0, 1] ...
... is devoted to showing that this sequence is interesting. In particular, we shall look at the following properties. • n 7→ an+1 /an is a bijection between the positive natural numbers and the positive rational numbers, • n/2k 7→ an /an+2k extends to a continuous strictly increasing function on [0, 1] ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.