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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, is
The nth row sums to n(n 2 + 1)/2, the constant of an n × n magic square (sequence A006003 in the OEIS). Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers, triangular numbers with an index that is triangular. [5]
In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if T n = n ( n + 1 ) / 2 {\displaystyle T_{n}=n(n+1)/2} denotes the n {\displaystyle n} th triangular number, then the doubly triangular numbers are the numbers of the form ...
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 case where one or both of the sequences is cubic is more difficult and as far as I know there's no general method to solve such problems. I looked specifically at triangular tetrahedral numbers (a triangular pyramid is usually called a tetrahedron). These are given in OEIS: A027568, which lists 6 of them, a surprising number given it's finite.
A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers. A truncated triangular pyramid number [1] is found by removing some smaller tetrahedral number (or triangular pyramidal number) from each of the vertices of a bigger tetrahedral number.
Angel numbers are repeating number sequences, often used as guides for deeper spiritual exploration. Ranging from 000 to 999 , each sequence carries its own distinct meaning and energy.
The sequence of squared triangular numbers is [4] 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 .
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