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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 ...
a number represented as a discrete r-dimensional regular geometric pattern of r-dimensional balls such as a polygonal number (for r = 2) or a polyhedral number (for r = 3). a member of the subset of the sets above containing only triangular numbers, pyramidal numbers , and their analogs in other dimensions.
The nth partial sum is given by a simple formula: = = (+). This equation was known to the Pythagoreans as early as the sixth century BCE. [5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even. An observation of tetrahedral numbers: Te 5 = Te 4 + Te 3 + Te 2 + Te 1; Numbers that are both triangular and tetrahedral must satisfy the binomial coefficient equation: = (+) = (+) =.
Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula = + = (+) / = where M p is a Mersenne prime. No odd perfect numbers are known, hence all known perfect ...
Consequently, a square number is also triangular if and only if + is square, that is, there are numbers and such that =. This is an instance of the Pell equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} .
The nth pentagonal number p n is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of ...
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 ...