Search results
Results from the WOW.Com Content Network
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.
Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the n th centered k-gonal number is obtained by the formula = + where T is a triangular number. The positive difference of two triangular numbers is a trapezoidal number.
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
Hexagonal number spiral with prime numbers in green and more highly composite numbers in darker shades of blue. Number spiral with 7503 primes visible on regular triangle. Ulam spiral with 10 million primes.
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.
In mathematics, a heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The n -th heptagonal number is given by the formula H n = 5 n 2 − 3 n 2 {\displaystyle H_{n}={\frac {5n^{2}-3n}{2}}} .