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Hofstadter Sequence
In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations. Sequences presented in ''Gödel, Escher, Bach: an Eternal Golden Braid'' The first Hofstadter sequences were described by Douglas Richard Hofstadter in his book ''Gödel, Escher, Bach''. In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are: Hofstadter Figure-Figure sequences The Hofstadter Figure-Figure (R and S) sequences are a pair of complementary integer sequences defined as follows: : \begin R(1) &= 1,\ S(1) = 2; \\ R(n) &= R(n - 1) + S(n - 1), \quad n > 1, \end with the sequence S(n) defined as a strictly increasing series of positive integers not present in R(n). The first few terms of these sequences are : ''R'': 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260,&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci number, Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the ''n''th perfect number. Computable and definable sequences An integer sequence is computable function, computable if th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange loops, artificial intelligence, and discovery in mathematics and physics. His 1979 book ''Gödel, Escher, Bach, Gödel, Escher, Bach: An Eternal Golden Braid'' won the Pulitzer Prize for general nonfiction,"General Nonfiction" . ''Past winners and finalists by category''. The Pulitzer Prizes. Retrieved 17 March 2012. and a National Book Award (at that time called The American Book Award) for Science."National Book Awards – 1980" [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel, Escher, Bach
''Gödel, Escher, Bach: an Eternal Golden Braid'' (abbreviated as ''GEB'') is a 1979 nonfiction book by American cognitive scientist Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. It also discusses self-reference and formal rules, isomorphism, what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself. In response to confusion over the book's theme, Hofstadter emphasized that ''Gödel, Escher, Bach'' is not about the relationships of mathematics and art, mathematics, art, and music and mathematics, music, but rather ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambek–Moser Theorem
The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two Complement (set theory), complementary sets. For instance, it applies to the partition of numbers into even number, even and odd number, odd, or into prime number, prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states that any two monotonic function, non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states that every complementary partition can be constructed in this way. When a formula is known for the natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the number not in the set. The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greg Huber
Greg is a masculine given name, and often a shortened form of the given name Gregory. Greg (sometimes spelled " Gregg") is also a surname. People with the name *Greg Abbott (other), multiple people *Greg Abel (born 1961/1962), Canadian businessman * Greg Adams (other), multiple people *Greg Allen (other), multiple people *Greg Anderson (other), multiple people * Greg Austin (other), multiple people *Greg Ball (other), multiple people *Greg Bell (other), multiple people *Greg Bennett (other), multiple people *Greg Berlanti (born 1972), American writer and producer *Greg Biffle (born 1969), American NASCAR driver * Greg Blankenship (born 1954), American football player *Greg Boyd (other), multiple people * Greg Boyer (other), multiple people *Greg Brady (broadcaster) (born 1971), Canadian sports radio host *Greg Brock (baseball) (born 1957), American baseball player * Greg Brooker (disambiguatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klaus Pinn
Klaus is a German, Dutch and Scandinavian given name and surname. It originated as a short form of Nikolaus, a German form of the Greek given name Nicholas. Notable persons whose family name is Klaus * Billy Klaus (1928–2006), American baseball player *Chris Klaus (born 1973), American entrepreneur *Felix Klaus (born 1992), German football player, son of Fred Klaus *Frank Klaus (1887–1948), German-American boxer, 1913 Middleweight Champion * Fred Klaus (born 1967), German football player and manager, father of Felix Klaus *Josef Klaus (1910–2001), Chancellor of Austria 1966–1970 *Karl Ernst Claus (1796–1864), Russian chemist *Václav Klaus (born 1941), Czech politician, former President of the Czech Republic * Walter K. Klaus (1912–2012), American politician and farmer Notable persons whose given name is Klaus * Brother Klaus, Swiss patron saint *Klaus Augenthaler (born 1957), German football player and manager *Klaus Badelt (born 1967), German composer *Klaus Ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
