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Later, it became a significant topic for Euler, who gave an explicit formula for all triangular numbers that are also perfect squares, among many other discoveries relating to figurate numbers. Figurate numbers have played a significant role in modern recreational mathematics. [9] In research mathematics, figurate numbers are studied by way of ...
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number. Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. For n > 2, the sum of the first n centered triangular numbers is the magic constant for an n ...
A triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the N th Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (70 2 × 2 2 = 140 2 = ) 19600. This is comparable with the 24th square pyramid having a total of 70 2 cannonballs. [5]
This category includes not only articles about certain types of figurate numbers, but also articles about theorems and conjectures pertaining to, and properties of, figurate numbers. Subcategories This category has only the following subcategory.
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.
For the case of triangular hexagonal numbers the ratio of a to d is a square and things work out a bit differently. In fact all hexagonal numbers are also triangular numbers. (This material could all be covered in an undergraduate level number theory course and it's conceivable that the corresponding problems could be assigned as homework.
Each layer represents one of the first five triangular numbers. 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 n th tetrahedral number, Te n, is the sum of the first n triangular numbers, that is,