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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Triangular number ： ウィキペディア英語版 Triangular number A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The ''n''th triangular number is the number of dots composing a triangle with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is: :0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 … The triangle numbers are given by the following explicit formulas: :$$ T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = where $\backslash textstyle$ is a binomial coefficient. It represents the number of distinct pairs that can be selected from ''n'' + 1 objects, and it is read aloud as "n plus one choose two". Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying n/2 pairs of numbers in the sum by the values of each pair n+1. However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The triangular number solves the "handshake problem" of counting the number of handshakes if each person in a room with ''n'' + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of ''n'' people is ''T''''n''−1.〔http://www.mathcircles.org/node/835〕 The function is the additive analog of the factorial function, which is the ''products'' of integers from 1 to ''n''. The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation: : $$ L_n = 3 T_= 3;~~~L_n = L_ + 3(n-1), ~L_1 = 0. In the limit, the ratio between the two numbers, dots and line segments is : $$ \lim_ \frac = \frac ==Relations to other figurate numbers== Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically, :$T\_n\; +\; T\_\; =\; \backslash left\; (\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right\; )\; =\; \backslash left\; (\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right\; )\; =\; n^2\; =\; (T\_n\; -\; T\_)^2.$ Alternatively, the same fact can be demonstrated graphically: There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36. Some of them can be generated by a simple recursive formula: :$S\_\; =\; 4S\_n\; \backslash left(\; 8S\_n\; +\; 1\backslash right)$ with $S\_1\; =\; 1.$ ''All'' square triangular numbers are found from the recursion :$S\_n\; =\; 34S\_\; -\; S\_\; +\; 2$ with $S\_0\; =\; 0$ and $S\_1\; =\; 1.$ Also, the square of the ''n''th triangular number is the same as the sum of the cubes of the integers 1 to ''n''. The sum of the all triangular numbers up to the ''n''th triangular number is the ''n''th tetrahedral number, :$\backslash frac\; .$ More generally, the difference between the ''n''th ''m''-gonal number and the ''n''th -gonal number is the th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number: the ''n''th centered ''k''-gonal number is obtained by the formula :$Ck\_n\; =\; kT\_+1\backslash$ where is a triangular number. The positive difference of two triangular numbers is a trapezoidal number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Triangular number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.