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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 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 dihedral angle of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°. [ 4 ] The truncated icosahedron is an Archimedean solid , meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in ...
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.