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Triangular Numbers
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 sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is 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 first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Six Triangular Numbers
First or 1st is the ordinal form of the number one (#1). First or 1st may also refer to: *World record, specifically the first instance of a particular achievement Arts and media Music * 1$T, American rapper, singer-songwriter, DJ, and record producer Albums * ''1st'' (album), a 1983 album by Streets * ''1st'' (Rasmus EP), a 1995 EP by The Rasmus, frequently identified as a single * '' 1ST'', a 2021 album by SixTones * ''First'' (Baroness EP), an EP by Baroness * ''First'' (Ferlyn G EP), an EP by Ferlyn G * ''First'' (David Gates album), an album by David Gates * ''First'' (O'Bryan album), an album by O'Bryan * ''First'' (Raymond Lam album), an album by Raymond Lam * ''First'', an album by Denise Ho Songs * "First" (Cold War Kids song), a song by Cold War Kids * "First" (Lindsay Lohan song), a song by Lindsay Lohan * "First", a song by Everglow from ''Last Melody'' * "First", a song by Lauren Daigle * "First", a song by Niki & Gabi * "First", a song by Jonas Broth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Polygonal Number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer. Examples Each centered ''k''-gonal number in the series is ''k'' times the previous triangular number, plus 1. This can be formalized by the expression \frac +1, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression \frac +1. These series consist of the *centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (), *centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Number
A hexagonal number is a figurate number. The ''n''th hexagonal number ''h''''n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the ''n''th hexagonal number :h_n= 2n^2-n = n(2n-1) = \frac. The first few hexagonal numbers are: : 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946... Every hexagonal number is a triangular number, but only every ''other'' triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula :M_p 2^ = M_p \frac = h_=h_ :where ''M''''p'' is a Mersenne prime. No odd perfect numbers are known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagonal Number
A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula :H_n=\frac. The first few heptagonal numbers are: : 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … Parity The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number. Additional properties * The heptagonal numbers have several notable formulas: :H_=H_m+H_n+5mn :H_=H_m+H_n-5mn+3n :H_m-H_n=\frac :40H_n+9=(10n-3)^2 Sum of reciprocals A formula for the sum of the reciprocals of the heptagonal numbers is given by: : \begin\sum_^\infty \frac &= \frac+\frac\ln(5)+\frac\ln\left(\frac\sqrt\right)+\frac\ln\left(\frac\sqrt\right)\\ &=\frac13\left(\frac+\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square (geometry), square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red. Triangular numbers : Square numbers : Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructe ... [...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 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, 35, 56, 84, 120, 165, 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 number is given by :T_n=\frac. It proceeds by induction. ;Base case :Te_1 = 1 = \frac. ;Inductive step :\ ... [...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 = \bigg(\sum_^n k\bigg)^2. This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). 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 cubes equals the sum of the first n(n+1)/2 odd numbers, that is, the odd numbers from 1 to n(n+1)-1. The average of these numbers is obviously n(n+1)/2, and there are n(n+1)/2 of them, so their sum is \bigl(n(n+1)/2\bigr)^2. Many early mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus Theorem 3D
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan). He was a Neopythagorean, who wrote about the mystical properties of numbers.Eric Temple Bell (1940), ''The development of mathematics'', page 83.Frank J. Swetz (2013), ''The European Mathematical Awakening'', page 17, Courier Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa.} Historians consider him a Neopythagorean based on his tendency to view numbers as having mystical properties. The age in which he lived (c. 100 AD) is only known because he mentions Thrasyllus in his ''Manual of Harmonics'', and because his ''Introduction to Arithmetic'' was apparently translated into Latin in the mid 2nd century by Apuleius.Henrietta ... [...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 perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Explicit formulas Write for the th square triangular number, and write and for the sides of the corresponding square and triangle, so that :N_k = s_k^2 = \frac. Define the ''triangular root'' of a triangular number to be . From this definition and the quadratic formula, :n = \frac. Therefore, is triangular ( is an integer) if and only if is square. Consequently, a square number is also triangular if and only if is square, that is, there are numbers and such that . This is an instance of the Pell equation with . All Pell equations have the trivial solution for any ; this is called the zeroth solution, and indexed as . If denotes the th nontrivial solution to any Pell equation for a particular , it can be shown by the method of desc ... [...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 few 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 … . 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 numbers alongside the triangular numbers and square numbers in Aristotle's ''Metaphysics'', and their discovery has been attributed much earlier to the Pythagoreans.. As a kind of figurate number, the pronic numbers are somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number 25 As Sum Of Two Triangular Numbers
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
