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: 1 , 3 , 6 , 10 , 15 , 21 , 28 , 36 , 45 , 55 , ... Since each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers. The formula for the ''n''th triangular number is ½''n''(''n'' + 1) or (1 + 2 + 3 + ... + {Link without Title} + {Link without Title} + ''n''). It is the Binomial Coefficient : It can also be shown that for any n-dimensional Simplex with sides of length x, the formula yields the number of points that make up the simplex. For example, a Tetrahedron with sides of length 2 corresponds to the number (2)(2 + 1)(2 + 2)/6, or 4. The four points forming this configuration are the Vertices of the tetrahedron. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangles = 3 plus 1 triangles = 1 = 4.) One of the most famous triangular numbers is 666 , also known as the Number Of The Beast . Every Even Perfect Number is triangular. The sum of two consecutive triangular numbers is a Square Number . This can be shown mathematically thus: the sum of the ''n''th and (''n''-1)th triangular numbers is {½''n''(''n'' + 1)} + {½(''n'' − 1)''n''}. This simplifies to (½''n''2 + ½''n'') + (½''n''2 − ½''n''), and thus to ''n''2. Alternatively, it can be demonstrated diagrammatically, thus: In each of the above examples, a square is formed from two interlocking triangles. More generally, the difference between the nth m -gonal Number and the nth (m+1)-gonal number is the (n-1)th triangular number. For example, the sixth Heptagonal Number (81) minus the sixth Hexagonal Number (66) equals the fifth triangular number, 15. Also, the square of a triangular number ''n'' is the same as the sum of the cubes of the integers 1 to ''n''. In Base 10 , the Digital Root of a triangular number is always 1, 3, 6 or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine: :6 = 3×2, :10 = 9×1+1, :15 = 3×5, :21 = 3×7, :28 = 9×3+1, :... Triangular numbers have all sorts of relations to other Figurate Numbers . Whenever a triangular number is divisible by 3, one third of it will be a Pentagonal Number . 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 where ''T'' is a triangular number. There are infinitely many triangular numbers that are also Square Number s; e.g., 1, 36. Some of them can be generated by a simple recursive formula: : with ''All'' Square Triangular Number s are found from the recursion : with and Two other interesting formulas regarding triangular numbers are: : and :, both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra. EXTERNAL LINKS
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