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Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid (geometry), pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, : Te_n = \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right) The tetrahedral numbers are: :1, 4, 10, 20 (number), 20, 35 (number), 35, 56 (number), 56, 84 (number), 84, 120 (number), 120, 165 (number), 165, 220 (number), 220, ... Formula The formula for the th tetrahedral number is represented by the 3rd rising factorial of divided by the factorial of 3: :Te_n= \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right)=\frac = \frac The tetrahedral numbers can also be represented as binomial coefficients: :Te_n=\binom. Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle. Proofs of formula This proof uses the fact that the th triangular num ... [...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] |
Proceedings Of The Royal Irish Academy, Section C
The ''Proceedings of the Royal Irish Academy'' (''PRIA'') is the journal of the Royal Irish Academy, founded in 1785 to promote the study of science, polite literature, and antiquities Antiquities are objects from antiquity, especially the civilizations of the Mediterranean such as the Classical antiquity of Greece and Rome, Ancient Egypt, and the other Ancient Near Eastern cultures such as Ancient Persia (Iran). Artifact .... It was known as several titles over the years: *1836–1866: ''Proceedings of the Royal Irish Academy'' *1870–1884: ''Proceedings of the Royal Irish Academy. Science'' *1879: ''Proceedings of the Royal Irish Academy. Polite Literature and Antiquities'' *1889–1901: ''Proceedings of the Royal Irish Academy'' In 1902, the journal split into three sections ''Section A: Mathematical and Physical Sciences'', ''Section B: Biological, Geological, and Chemical Science'' and ''Section C: Archaeology, Culture, History, Literature''. ''Section A'' is now p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squared Triangular Number
In number theory, the sum of the first cubes is the square of the th triangular number. That is, :1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2. The same equation may be written more compactly using the mathematical notation for summation: :\sum_^n k^3 = \left(\sum_^n k\right)^2. This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa ( – ). History Nicomachus, at the end of Chapter 20 of his ''Introduction to Arithmetic'', pointed out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the first n cubes equals the sum of the first \tfrac odd numbers, that is, the odd numbers from 1 to n(n+1)-1. The average of these numbers is obviously \tfrac, and there are \tfrac of them, so their sum is \left(\tfrac\right)^2. Many early mathematicians have stud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus Theorem 3D
Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'', which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education. Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa. His ''Manual of Harmonics'' was addressed to a lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected scholar o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pronic Number
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers. The first 60 pronic numbers are: : 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... . Letting P_n denote the pronic number n(n+1), we have P_ = P_. Therefore, in discussing pronic numbers, we may assume that n\geq 0 without loss of generality, a convention that is adopted in the following sections. As figurate numbers The pronic numbers were studied as figurate nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number 25 As Sum Of Two Triangular Numbers
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge, now ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number 16 As Sum Of Two Triangular Numbers
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degree (angle), degrees, or Pi, /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called square (algebra), squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digit Sum
In mathematics, the digit sum of a natural number in a given radix, number base is the sum of all its numerical digit, digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. We define the digit sum for base b > 1, F_b : \mathbb \rightarrow \mathbb to be the following: :F_b(n) = \sum_^ d_i where k = \lfloor \log_ \rfloor is one less than the number of digits in the number in base b, and :d_i = \frac is the value of each digit of the number. For example, in base 10, the digit sum of 84001 is F_(84001) = 8 + 4 + 0 + 0 + 1 = 13. For any two bases 2 \leq b_1 < b_2 and for sufficiently large natural numbers :. The sum of the base 10 digits of the integers 0, 1, 2, ... is given by in the On-Line Encyclopedia of Integer Sequences. use the generating function of this integer sequence (and of the analogous sequence for binary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theon Of Smyrna
Theon of Smyrna ( ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. His surviving ''On Mathematics Useful for the Understanding of Plato'' is an introductory survey of Greek mathematics. Life Little is known about the life of Theon of Smyrna. A bust created at his death, and dedicated by his son, was discovered at Smyrna, and art historians date it to around 135 CE. Ptolemy refers several times in his ''Almagest'' to a Theon who made observations at Alexandria, but it is uncertain whether he is referring to Theon of Smyrna.James Evans, (1998), ''The History and Practice of Ancient Astronomy'', New York, Oxford University Press, 1998, p. 49 The lunar impact crater Theon Senior is named for him. Works Theon wrote several commentaries on the works of mathematicians and philosophers of the time, including works on the philosophy of Plato. Most of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division (mathematics)
Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the ''dividend'', which is divided by the ''divisor'', and the result is called the ''quotient''. At an elementary level the division of two natural numbers is, among other Quotition and partition, possible interpretations, the process of calculating the number of times one number is contained within another. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). However, this number of times or the number contained (divisor) need not be integers. The division with remainder or Euclidean division of two natural numbers provides an integer ''quotient'', which is the number of times the second number is completely contained in the first number, and a ''remainder'', which is the part of the first number that remains, when in the course of computing the quotient, no further ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary OR
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the processor. Most bitwise operations are presented as two-operand instructions where the result replaces one of the input operands. On simple low-cost processors, typically, bitwise operations are substantially faster than division, several times faster than multiplication, and sometimes significantly faster than addition. While modern processors usually perform addition and multiplication just as fast as bitwise operations due to their longer instruction pipelines and other architectural design choices, bitwise operations do commonly use less power because of the reduced use of resources. Bitwise operators In the explanations below, any indication of a bit's position is counted from the right (least signif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
