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The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a ...
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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 ...
A positive integer n for which there is an e > 0 such that d(n) / n e ≥ d(k) / k e for all k > 1. A002201: Pronic numbers: 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, ... a(n) = 2t(n) = n(n + 1), with n ≥ 0 where t(n) are the triangular numbers. A002378: Markov numbers: 1, 2, 5, 13, 29, 34, 89, 169, 194, ... Positive integer ...
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 ...
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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. The number to be removed may be same or different from each of the vertices. [2]
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 n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
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