A hexagonal number is a figurate number, The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side, as shown in [1]. A figurate number is a number that can be represented as a regular and discrete geometric pattern (e
The formula for the nth hexagonal number is:
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 not every triangular number is a hexagonal number. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Six is the second smallest Composite number, its proper Divisors being 1, 2 and 3. 28 ( twenty-eight) is the Natural number following 27 and preceding 29. 45 ( forty-five) is the Natural number following 44 and followed by 46. 66 ( sixty-six) is the Natural number following 65 and preceding 67. 91 ( ninety-one) is the Natural number following 90 and preceding 92. 190 is the natural number following one hundred [and] eighty-nine and preceding one hundred [and] ninety-one. Four hundred ninety-six is the Natural number following four hundred ninety-five and preceding four hundred ninety-seven 500 ( five hundred) is the Natural number following 499 and preceding 501. A triangular number is the sum of the n Natural numbers from 1 to n. 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 (also repeated digital sum) of a number is the number obtained by adding all the digits then adding the digits of that number and then continuing until a single-digit
Any integer greater than 1791 can be expressed as a sum of at most four hexagonal numbers, a fact proven by Adrien-Marie Legendre in 1830. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. For the game see 1830 (board game. Year 1830 ( MDCCCXXX) was a Common year starting on Friday (link will display
Hexagonal numbers can be rearranged into rectangular numbers n long and 2n−1 tall (or vice versa).
Hexagonal numbers should not be confused with centered hexagonal numbers, which model the standard packaging of Vienna sausages. A centered hexagonal number, or hex number, is a centered Figurate number that represents a Hexagon with a dot in the center and all other dots A Vienna sausage is a wiener. The word wiener means Viennese in German. To avoid ambiguity, hexagonal numbers are sometimes called "cornered hexagonal numbers".
Test for hexagonal numbers
One can efficiently test whether a positive integer x is an hexagonal number by computing
If n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not hexagonal.
External links
- Mathworld entry on Hexagonal Number
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