Automatic sequence. An automatic sequence (or k-automatic sequence) is an infinite sequence of terms characterized by a finite automaton. The n-th term of the sequence is a mapping of the final state of the automaton when its input is the digits of n in some fixed base k.[1][2] A k-automatic set is a set of non-negative integers for which the sequence of values of its characteristic function is an automatic sequence: that is, membership of n in the set can be determined by a finite state automaton on the digits of n in base k.[3][4] An automaton reading the base k digits from the most significant is said to be direct reading, and from the least significant is reverse reading.[4] However the two directions lead to the same class of sequences.[5] Every automatic sequence is a morphic word.[6] Automaton point of view[edit] Let k be a positive integer, and D = (E, φ, e) be a deterministic automaton where Define a function m from the set of positive integers to the set A as follows: Substitution point of view[edit]
The Thue-Morse Sequence: A Turtle Graphics. Thue–Morse sequence. This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence. 5 logical matrices that give the beginning of the T. -M. sequence, when read line by line Either in set A (vertical index) or in set B (horizontal index) is an odd number of elements. In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. This procedure yields 0 then 01, 0110, 01101001, 0110100110010110, and so on. The infinite sequence begins: 01101001100101101001011001101001.... Any other ordered pair of symbols may be used instead of 0 and 1; the logical structure of the Thue–Morse sequence does not depend on the symbols that are used to represent it. Definition[edit] There are several equivalent ways of defining the Thue–Morse sequence.
Direct definition[edit] Recurrence relation[edit] L-system[edit] So with. 3 Ways to Generate the Thue Morse Sequence. Look-and-say sequence. The lines show the growth of the numbers of digits in the look-and-say sequences with starting points 23 (red), 1 (blue), 13 (violet), 312 (green). These lines (when represented in a logarithmic scale) tend to straight lines whose slopes coincide with Conway's constant.
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (sequence A005150 in OEIS). To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example: 1 is read off as "one 1" or 11.11 is read off as "two 1s" or 21.21 is read off as "one 2, then one 1" or 1211.1211 is read off as "one 1, then one 2, then two 1s" or 111221.111221 is read off as "three 1s, then two 2s, then one 1" or 312211.
The idea of the look-and-say sequence is similar to that of run-length encoding. d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, … Feigenbaum constants. In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell Feigenbaum. History[edit] The first constant[edit] The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map where f(x) is a function parameterized by the bifurcation parameter a. It is given by the limit:[2] where an are discrete values of a at the nth period doubling. According to (sequence A006890 in OEIS), this number to 30 decimal places is: δ = 4.669 201 609 102 990 671 853 203 821 578(...). Illustration[edit] Non-linear maps[edit] To see how this number arises, consider the real one-parameter map: Here a is the bifurcation parameter, x is the variable.
The ratio in the last column converges to the first Feigenbaum constant. With real parameter a and variable x. Fractals[edit] and.