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- Triangular_number abstract "A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers (sequence A000217 in OEIS), starting at the 0th triangular number, is:0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120 …The triangle numbers are given by the following explicit formulas:where is a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two". Carl Friedrich Gauss is said to have discovered this relationship early on in his youth, though the legitimacy of such a claim has been called into question.The triangular number Tn solves the "handshake problem" of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is Tn−1.Triangle numbers are the additive analog of the factorials, which are the products of integers from 1 to n.The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation: In the limit, the ratio between the two numbers, dots and line segments is".
- Triangular_number thumbnail First_six_triangular_numbers.svg?width=300.
- Triangular_number wikiPageExternalLink hypertetrahedral-polytopic-roots.
- Triangular_number wikiPageExternalLink gauss.
- Triangular_number wikiPageExternalLink square.
- Triangular_number wikiPageExternalLink triSquare.shtml.
- Triangular_number wikiPageID "213915".
- Triangular_number wikiPageRevisionID "604666845".
- Triangular_number hasPhotoCollection Triangular_number.
- Triangular_number id "p/a013370".
- Triangular_number title "Arithmetic series".
- Triangular_number title "Triangular Number".
- Triangular_number urlname "TriangularNumber".
- Triangular_number subject Category:Figurate_numbers.
- Triangular_number subject Category:Integer_sequences.
- Triangular_number subject Category:Proof_without_words.
- Triangular_number subject Category:Triangles.
- Triangular_number type Abstraction100002137.
- Triangular_number type Amount105107765.
- Triangular_number type Arrangement107938773.
- Triangular_number type Attribute100024264.
- Triangular_number type FigurateNumbers.
- Triangular_number type Figure113862780.
- Triangular_number type Group100031264.
- Triangular_number type IntegerSequences.
- Triangular_number type Magnitude105090441.
- Triangular_number type Number105121418.
- Triangular_number type Ordering108456993.
- Triangular_number type PlaneFigure113863186.
- Triangular_number type Polygon113866144.
- Triangular_number type Property104916342.
- Triangular_number type Sequence108459252.
- Triangular_number type Series108457976.
- Triangular_number type Shape100027807.
- Triangular_number type Triangle113879320.
- Triangular_number type Triangles.
- Triangular_number comment "A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.".
- Triangular_number label "Dreieckszahl".
- Triangular_number label "Driehoeksgetal".
- Triangular_number label "Liczba trójkątna".
- Triangular_number label "Nombre triangulaire".
- Triangular_number label "Numero triangolare".
- Triangular_number label "Número triangular".
- Triangular_number label "Número triangular".
- Triangular_number label "Triangular number".
- Triangular_number label "Треугольное число".
- Triangular_number label "عدد مثلثي".
- Triangular_number label "三角形數".
- Triangular_number label "三角数".
- Triangular_number sameAs Trojúhelníkové_číslo.
- Triangular_number sameAs Dreieckszahl.
- Triangular_number sameAs Número_triangular.
- Triangular_number sameAs Nombre_triangulaire.
- Triangular_number sameAs Numero_triangolare.
- Triangular_number sameAs 三角数.
- Triangular_number sameAs 삼각수.
- Triangular_number sameAs Driehoeksgetal.
- Triangular_number sameAs Liczba_trójkątna.
- Triangular_number sameAs Número_triangular.
- Triangular_number sameAs m.01fbc7.
- Triangular_number sameAs Q245102.
- Triangular_number sameAs Q245102.
- Triangular_number sameAs Triangular_number.
- Triangular_number wasDerivedFrom Triangular_number?oldid=604666845.
- Triangular_number depiction First_six_triangular_numbers.svg.
- Triangular_number isPrimaryTopicOf Triangular_number.
