Question: Information: - A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The th triangular number is the number of dots composing a triangle with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is - In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. - The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean - A hexagonal number is a figurate number . The nth hexagonal number hn 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 formula for the nth hexagonal number h_n = 2n ^ 2 - n = n ( 2n - 1 ) = ( ( 2n ) \ times ( ( 2n - 1 ) ) \ over 2 ) . \ , \ ! The first few hexagonal numbers ( sequence A000384 in OEIS ) are : 1 , 6 , 15 , 28 , 45 , 66 , 91 , 120 , 153 , 190 , 231 , 276 , 325 , 378 , 435 , 496 , 561 , 630 , 703 , 780 , 861 , 946 . 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 is hexagonal , given by the formula M_p 2 ^ ( p - 1 ) = M_p ( M_p + 1 ) / 2 = h -- ( ( M_p +1 ) / 2 ) = h -- ( 2 ^ ( p - 1 ) ) where Mp is a Mersenne prime . No odd perfect numbers are known , hence all known perfect numbers are hexagonal . For example , the 2nd hexagonal number is 2 3 = 6 ; the 4th is 4 7 = 28 ; the 16th is 16 31 = 496 ; and the 64th is 64 127 = 8128 . The largest number that can not be written as a sum of at most four hexagonal numbers is 130 . Adrien - Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way . Hexagonal numbers can be rearranged into rectangular numbers of size n by ( 2n 1 ) . Hexagonal numbers should not be confused with centered hexagonal numbers , which model the standard packaging of Vienna sausages . To avoid ambiguity , hexagonal numbers are sometimes called '' cornered hexagonal numbers '' . Given the information above, choose from the list below the object entity that exhibits the relation 'subclass of' with the subject 'hexagonal number'. Choices: - figurate number - mathematics - natural - number - polygon - polygonal number - sequence - sum - term - triangle
Answer:
polygonal number