In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2...

mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words...

number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on...

squared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with...

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns...

properties of oblong, triangular, and square numbers: 1 . The number 10 for example, can be arranged as a triangle (see triangular number): But 10 cannot be...

non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than...

152), an octagonal number, and a squared triangular number (225 = (1 + 2 + 3 + 4 + 5)2 = 13 + 23 + 33 + 43 + 53) . As the square of a double factorial...

As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations...

A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other...

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. The term often refers to square pyramidal numbers...

multiplication table gives the Nicomachus theorem, proving that each squared triangular number is a sum of cubes. In an acute isosceles triangle, it is possible...

{\displaystyle 1^{3}+2^{3}+\cdots +n^{3}=(1+2+\cdots +n)^{2}} (see squared triangular number) Aryabhata's system of astronomy was called the audAyaka system...

n} th triangular number, then the doubly triangular numbers are the numbers of the form T T n {\displaystyle T_{T_{n}}} . The doubly triangular numbers...

007 : smallest prime number with 10 digits. 1,000,006,281 : smallest triangular number with 10 digits and the 44,721st triangular number. 1,000,014,129 =...

7-digit prime number 1,000,405 = Smallest triangular number with 7 digits and the 1,414th triangular number 1,002,001 = 10012, palindromic square 1,006,301...

n-th square pyramidal number. The number of rectangles in a square grid is given by the squared triangular numbers. The square pyramidal number P n {\displaystyle...

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 term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes...

Augmented triangular prism: The result of gluing a triangular prism to a square pyramid, this has six equilateral triangle faces and two square faces. It...

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center...

are both tetrahedral and square pyramidal. Square triangular number, the numbers that are simultaneously square and triangular Close-packing of equal spheres...

prime number 10,001,628 = smallest triangular number with 8 digits and the 4,472nd triangular number 10,004,569 = 31632, the smallest 8-digit square 10,077...

triangular number and n more than the nth square number, as given by the alternative formula n2 + n for pronic numbers. Hence the nth pronic number and...

initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the...

centered square number, Mertens function zero 1014 = 210-10, Mertens function zero, sum of the nontriangular numbers between successive triangular numbers...

is an 8 × 8 square. Its square, 2042 = 41616, is the fourth square triangular number. As a figurate number, 204 is also a nonagonal number and a truncated...

of 100 and also the square of 100,000. 10,000,000,019 = smallest 11-digit prime number. 10,000,020,331 = smallest triangular number with 11 digits and...

A nonagonal number, or an enneagonal number, is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided...

are square, then the right triangular prism is semiregular. A semiregular prism means that the number of its polygonal base's edges equals the number of...