6/12/2023 0 Comments Recursive sequence![]() Zooming in on other portions of the set yields fascinating swirling shapes. The Mandelbrot set, for having such a simple definition, exhibits immense complexity. Greens, reds, and purples can be seen when we zoom in – those are used for numbers that grow very slowly. For example, in the image below, light blue is used for numbers that get large quickly, while darker shades are used for numbers that grow more slowly. What is a recursion A recursion is a list of values, where later values are built from earlier values. Numbers that get big fast are colored one shade, while colors that are slow to grow are colored another shade. For example, using \(c=1 i\) above, the sequence was distance 2 from the origin after only two recursions.įor some other numbers, it may take tens or hundreds of iterations for the sequence to get far from the origin. To create a meaningful coloring, often people count the number of iterations of the recursive sequence that are required for a point to get further than 2 units away from the origin. In addition to coloring the Mandelbrot set itself black, it is common to the color the points in the complex plane surrounding the set. The boundary of this shape exhibits quasi-self-similarity, in that portions look very similar to the whole. If all complex numbers are tested, and we plot each number that is in the Mandelbrot set on the complex plane, we obtain the shape to the right. The goal of this paper is to study the expressive power of polynomial recursive sequences. 0.5 i=-0.02734 0.40625 i\) Footnote 1 Moreover, over unary alphabets they define exactly polynomial recursive sequences, in the same fashion as weighted automata (respectively order 2 pushdown automata) over unary alphabets define linear recursive sequences. If we go with that definition of a recursive sequence, then both arithmetic sequences and geometric sequences are also recursive. Surprisingly, these three models, although introduced in different contexts, are all equivalent. Lets explore the two phases of solving recursive sequences: Phase I: Re-subsitute values into f ( x) until you reach the 'seed value' (in programming its often called the 'base. ![]() Polynomial automata, connected to reachability problems for vector addition systems. Polynomial recurrent relations that generalise pushdown automata of order 3 Ĭost-register automata which arose as a variant of streaming transducers Thus, nonlinear extensions of linear recursive sequences may correspond to nonlinear extensions of weighted automata. ![]() Pushdown automata of order k can be used for defining mappings from words to words in particular, for k = 2 and 1-letter alphabets, such automata compute exactly the linear recursive sequences of natural integers. It is known that sequences definable in this way by weighted automata are exactly the linear recursive sequences. Then a weighted automaton defines a mapping from natural numbers (possible lengths) to rationals, and this can be seen as a sequence. In the special case of a 1-letter alphabet, each word can be identified with its length. Lets find a recursive formula for the sequence. Weighted automata over the rational semiring are a quantitative variant of finite automata that assign rational numbers to words. The two classes of linear and polynomial recursive sequences appear naturally in automata theory, and in particular in connection with weighted automata and higher-order pushdown automata. Thus, the recurrence relation uses two polynomials: P 1( x 1, x 2) = x 1 x 2 and P 2( x 1, x 2) = x 2 1.
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