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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 ...
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 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 ...
These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong, triangular, and square numbers [1]: 1 .
This category includes not only articles about certain types of figurate numbers, ... Centered tetrahedral number; Centered triangular number; Cube (algebra) D.
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 pyramid with side length 5 contains 35 spheres. 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.
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]
These numbers can be viewed as figurate numbers, a four-dimensional hyperpyramidal generalization of the triangular numbers and square pyramidal numbers. As Stein (1971) observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an grid.