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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 ...
Centered hexagonal numbers appearing in the Catan board game: 19 land tiles, 37 total tiles. In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, [1] [2] is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice.
This is exactly the general form of a centered hexagonal number; that is, all of these cuban primes are centered hexagonal. As of July 2023 [update] the largest known has 3,153,105 digits with y = 3 3304301 − 1 {\displaystyle y=3^{3304301}-1} , [ 2 ] found by R.Propper and S.Batalov.
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.
A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant M. A normal magic hexagon contains the consecutive integers from 1 to 3n 2 − 3n + 1.
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.
This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to . The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of ...
A centered heptagonal prime is a centered heptagonal number that is prime. The first few centered heptagonal primes are The first few centered heptagonal primes are 43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, ...
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