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All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [4]A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the th triangular number (+) is square, then so is the larger (+) th triangular number, since:
The sequence of squared triangular numbers is [1] 0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, ... . 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 ...
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 ...
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.
The left side of this identity describes a square number, while the right side describes a triangular number, so the result is a square triangular number. Falcón and Díaz-Barrero (2006) proved another identity relating Pell numbers to squares and showing that the sum of the Pell numbers up to P 4n +1 is always a square:
A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. [1] The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides. [2] The numbers of points in the base and in layers parallel to the base are given by ...
Palindromic number. A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word (such as rotor or ...
Viète. de Moivre. Euler. Fourier. v. t. e. In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.