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Factoriangular Number
In number theory, a factoriangular number is an integer formed by adding a factorial and a triangular number with the same index. The name is a portmanteau of "factorial" and "triangular." Definition For n \ge 1, the nth ''factoriangular number'', denoted \operatorname_n, is defined as the sum of the nth factorial and the nth triangular number: :\operatorname_n = n! + T_n = n! + \frac. The first few factoriangular numbers are: These numbers form the integer sequencebr>A101292in the Online Encyclopedia of Integer Sequences (OEIS). Properties Recurrence relations Factoriangular numbers satisfy several recurrence relations. For n \geq 1, :\operatorname_ = (n+1)\left(\operatorname_n - \frac\right) And for n \geq 2, :\operatorname_n = n\left(\operatorname_ - \frac\right) These are linear non-homogeneous recurrence relations with variable coefficients of order 1. Generating functions The exponential generating function E(x) = \sum_^\infty \operatorname_n \tfrac for facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
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] |
Square Triangular Number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the sum of all integers from 1 to n has a square root that is an integer. There are infinitely many square triangular numbers; the first few are: Solution as a Pell equation Write N_k for the kth square triangular number, and write s_k and t_k for the sides of the corresponding square and triangle, so that Define the ''triangular root'' of a triangular number N=\tfrac to be n. From this definition and the quadratic formula, Therefore, N is triangular (n is an integer) if and only if 8N+1 is square. Consequently, a square number M^2 is also triangular if and only if 8M^2+1 is square, that is, there are numbers x and y such that x^2-8y^2=1. This is an instance of the Pell equation x^2-ny^2=1 with n=8. All Pell equations have the trivial solution x=1,y=0 for any n; this is called the zeroth solution, and indexed as (x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazy Caterer's Sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a Disk (mathematics), disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point inside the circle, but up to seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines; for generalizations to higher dimensions, ''see'' arrangement of hyperplanes. The analogue of this integer sequence, sequence in three dimensions is the cake numbers. Formula and sequence The maximum number ''p'' of pieces that can be created with a given number of cuts (where ) is given by the formula :p = \frac. Using binomial coefficients, the formula can be expressed as :p = 1 + \dbinom = \dbinom+\dbinom+\dbinom. Simply put, each number equals a triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Prime
A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for ''n'' = 1, 2, 3, 4, 6, 7, 11, 12, 14) are : : 2 (0! + 1 or 1! + 1), 3 (2! + 1), 5 (3! − 1), 7 (3! + 1), 23 (4! − 1), 719 (6! − 1), 5039 (7! − 1), 39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1), ... ''n''! − 1 is prime for : :''n'' = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003, 632760, ... (resulting in 28 factorial primes) ''n''! + 1 is prime for : :''n'' = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, 288465, 308084, 422429, ... (resulting in 24 factorial primes - the prime 2 is repeated) No other factori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Triangular Number
In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if T_n=n(n+1)/2 denotes the nth triangular number, then the doubly triangular numbers are the numbers of the form T_. Sequence and formula The doubly triangular numbers form the sequence :0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, ... The nth doubly triangular number is given by the quartic formula T_ = \frac. The sums of row sums of Floyd's triangle give the doubly triangular numbers. Another way of expressing this fact is that the sum of all of the numbers in the first n rows of Floyd's triangle is the nth doubly triangular number. Sum of reciprocals A formula for the sum of the reciprocals of the doubly triangular numbers is given by \sum_^\frac=\sum_^\frac=6-\frac\tanh\left(\frac\right). In combinatorial enumeration Doubly triangular numbers arise naturally as numbers of of objects, inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the central binomial coefficients by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : . Properties An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a #Second proof, proof of the correctness of the formula. Another alternative expression is :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell Number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , and , so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + . As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indagationes Mathematicae
''Indagationes Mathematicae'' (Latin for "Mathematical Investigations") is a Dutch mathematics journal. The journal originates from the ''Proceedings of the Royal Netherlands Academy of Arts and Sciences'' (or ''Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen''), founded in 1895. From 1939, mathematics articles in this journal were published separately, under the alternative title ''Indagationes Mathematicae''. In 1951 the proceedings officially split into three journals, keeping the same name but distinguished from each other by being in separate series. They were Series A (Mathematical Sciences), Series B (Physical Sciences), and Series C (Biological and Medical Sciences). At that time, Series A became published by the North-Holland Publishing Company; the volumes from this time are now listed by the publisher as ''Indagationes Mathematicae (Proceedings)''. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factorization
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number. For example, is a composite number because , but is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example . Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers , , , and so on, up to the square root of . For larger numbers, especially when using a computer, various more sophis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
