Hexagonal Numbers : Means, Formula, Properties and Test

Hexagonal numbers are a unique class of figurate numbers (Figurate numbers are numbers that can be represented by dots arranged in geometric shapes like triangle, square, hexagon, etc.) Hexagonal numbers belong to this family, specifically forming hexagons that can be represented as points arranged in the shape of a regular hexagon.

First Few Hexagonal Numbers are:

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560, . . .

Here is the visual representation of the hexagonal numbers from H₁ to H₄:

Formula of Hexagonal Number

The formula to find the n^th hexagonal number is:

H_n = 2(n²) - n

In this formula:

• H_n represents the n^th hexagonal number.
• n represents the position in the sequence of hexagonal numbers (starting from 1).

Here is a table of hexagonal numbers up to H_10:

Note: First few hexagonal numbers which are perfect square are: 1, 1225, 1413721, 1631432881, 1882672131025, . . .

Properties of Hexagonal Numbers

Some properties of hexagonal numbers are:

• Hexagonal numbers can be arranged in hexagon shapes.
• Every hexagonal number is a triangular number.
• Every even perfect numbers is a hexagonal number.
• You can efficiently determine if a positive integer x is a hexagonal number by calculating: n = \frac{\sqrt{8x + 1} + 1}{4}.

Test for Hexagonal Numbers

To test if a given number is a hexagonal number, you can use the following formula:

n = \frac{(2k-1)k}{2}

where n is the hexagonal number and k is the position of the number in the sequence (1^st, 2^nd, 3^rd, etc.).

Let’s test if 45 is a hexagonal number:

k = \frac{1 + \sqrt{1 + 8 \times 45}}{4} = \frac{1 + \sqrt{361}}{4} = \frac{1 + 19}{4} = 5

Since k = 5 is a whole number, 45 is a hexagonal number.

Related Reads,

• Hexagon
• Centered Hexagonal Number
• Triangular Numbers