For triangular numbers that are themselves square, see square triangular number.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010). The nth coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the nth region is n times n x n.
In number theory, the sum of the first n cubes is the square of the nth triangular number. That is,
The same equation may be written more compactly using the mathematical notation for summation:
This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE).
and 20 Related for: Squared triangular number information
In number theory, the sum of the first n cubes is the square of the nth triangularnumber. That is, 1 3 + 2 3 + 3 3 + ⋯ + n 3 = ( 1 + 2 + 3 + ⋯ + n ) 2...
mathematics, a squaretriangularnumber (or triangularsquarenumber) is a number which is both a triangularnumber and a squarenumber. There are infinitely...
number, other examples being square numbers and cube numbers. The nth triangularnumber is the number of dots in the triangular arrangement with n dots on...
squared". The name squarenumber 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...
numbers. The number 10 for example, can be arranged as a triangle (see triangularnumber): But 10 cannot be arranged as a square. The number 9, on the other...
As well as being used to approximate the square root of two, Pell numbers can be used to find squaretriangular numbers, to construct integer approximations...
A centered (or centred) triangularnumber 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...
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 nth square pyramidal number. The number of rectangles in a square grid is given by the squaredtriangular numbers. The square pyramidal number P n {\displaystyle...
right triangular prism is semiregular. A semiregular prism means that the number of its polygonal base's edges equals the number of its square faces....
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes...
{\displaystyle 1^{3}+2^{3}+\cdots +n^{3}=(1+2+\cdots +n)^{2}} (see squaredtriangularnumber) Aryabhata's system of astronomy was called the audAyaka system...
also be a triangularnumber. A 10 × 10 {\displaystyle 10\times 10} magic square has a magic constant of 505, where this is the ninth number to have a...
n} th triangularnumber, then the doubly triangular numbers are the numbers of the form T T n {\displaystyle T_{T_{n}}} . The doubly triangular numbers...
the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming. No Fibonacci number can be a perfect number. More generally...
In elementary number theory, a centered squarenumber 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. Squaretriangularnumber, the numbers that are simultaneously square and triangular Close-packing of equal spheres...
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...
such representations of the number 17, for example, are shown below: 17 = 10 + 6 + 1 (triangular numbers) 17 = 16 + 1 (square numbers) 17 = 12 + 5 (pentagonal...
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