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Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number ...
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.
Download as PDF; Printable version; ... have mostly hexagonal faces. ... The triangulation number is T = b 2 + bc + c 2. This number times the number of original ...
a number represented as a discrete r-dimensional regular geometric pattern of r-dimensional balls such as a polygonal number (for r = 2) or a polyhedral number (for r = 3). a member of the subset of the sets above containing only triangular numbers, pyramidal numbers , and their analogs in other dimensions.
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.
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 ...
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}}} .
The n-th centered k-gonal number is equal to the n-th regular k-gonal number plus (n-1) 2. Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:
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