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A hexagonal number is a figurate number. The nth hexagonal number h n is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The first four hexagonal numbers. The formula for the nth hexagonal number = = = (). The ...
In the opposite direction, the index n corresponding to the centered hexagonal number = can be calculated using the formula n = 3 + 12 H − 3 6 . {\displaystyle n={\frac {3+{\sqrt {12H-3}}}{6}}.} This can be used as a test for whether a number H is centered hexagonal: it will be if and only if the above expression is an integer.
Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers. Every even perfect number is triangular (as well as hexagonal), given by the formula = (+) = where M p is a Mersenne prime. No odd perfect numbers are known; hence, all known perfect numbers are triangular.
A computer search for pentagonal square triangular numbers has yielded only the trivial value of 1, though a proof that there are no other such numbers has yet to be found. [5] The number 1225 is hecatonicositetragonal (s = 124), hexacontagonal (s = 60), icosienneagonal (s = 29), hexagonal, square, and triangular.
The number x is pentagonal if and only if n is a natural number. In that case x is the nth pentagonal number. For generalized pentagonal numbers, it is sufficient to just check if 24x + 1 is a perfect square. For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if
By the prime number theorem, this formula with A set equal to one is the asymptotic number of primes ... Hexagonal number spiral with prime numbers in green and more ...
Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles. The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem.
Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30. 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]