centered hexagonal number

Image:Centered hexagonal = 1 + 6triangular.svgs of {{math|n(n−1)}} dots each.]]

The {{mvar|n}}th centered hexagonal number is given by the formula

:$H(n)\; =\; n^3\; -\; (n-1)^3\; =\; 3n(n-1)+1\; =\; 3n^2\; -\; 3n\; +1.\; \backslash ,$

Expressing the formula as

:$H(n)\; =\; 1+6\backslash left(\backslash frac\{n(n-1)\}\{2\}\backslash right)$

shows that the centered hexagonal number for {{mvar|n}} is 1 more than 6 times the {{math|(n − 1)}}th triangular number.

In the opposite direction, the index {{mvar|n}} corresponding to the centered hexagonal number $H\; =\; H(n)$ can be calculated using the formula

:$n=\backslash frac\{3+\backslash sqrt\{12H-3\}\}\{6\}.$

This can be used as a test for whether a number {{mvar|H}} is centered hexagonal: it will be if and only if the above expression is an integer.

The centered hexagonal numbers $H(n)$ satisfy the recurrence relation

:$H(n+1)\; =\; H(n)\; +\; 6n.$

From this we can calculate the generating function $F(x)\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; H(n)\; x^n$. The generating function satisfies

:$F(x)\; =\; x\; +\; xF(x)\; +\; \backslash sum\_\{n\; \backslash ge\; 2\}\; 6n\; x^n.$

The latter term is the Taylor series of $\backslash frac\{6x\}\{(1-x)^2\}\; -\; 6x$, so we get

:$(1\; -\; x)\; F(x)\; =\; x\; +\; \backslash frac\{6x\}\{(1-x)^2\}\; -\; 6x\; =\; \backslash frac\{x\; +\; 4x^2\; +\; x^3\}\{(1-x)^2\}$

and end up at

:$F(x)\; =\; \backslash frac\{x\; +\; 4x^2\; +\; x^3\}\{(1-x)^3\}.$

File:visual_proof_centered_hexagonal_numbers_sum.svg of the sum of the first n hex numbers by arranging n³ semitransparent balls in a cube and viewing along a space diagonal – colour denotes cube layer and line style denotes hex number]]

In base 10 one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with period 5).

This follows from the last digit of the triangle numbers {{OEIS|id=A008954}} which repeat 0-1-3-1-0 when taken modulo 5.

In base 6 the rightmost digit is always 1: 1₆, 11₆, 31₆, 101₆, 141₆, 231₆, 331₆, 441₆...

This follows from the fact that every centered hexagonal number modulo 6 (=10₆) equals 1.

The sum of the first {{mvar|n}} centered hexagonal numbers is {{math|n³}}. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, but they represent different shapes. Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. (This can be seen geometrically from the diagram.) In particular, prime centered hexagonal numbers are cuban primes.

The difference between {{math|(2n)²}} and the {{mvar|n}}th centered hexagonal number is a number of the form {{math|3n² + 3n − 1}}, while the difference between {{math|(2n − 1)²}} and the {{mvar|n}}th centered hexagonal number is a pronic number.

File:comparison_optical_telescope_primary_mirrors.svg telescopes are centered hexagonal numbers]]

Many segmented mirror reflecting telescopes have primary mirrors comprising a centered hexagonal number of segments (neglecting the central segment removed to allow passage of light) to simplify the control system.Mast, T. S. and Nelson, J. E. [http://osti.gov/servlets/purl/6194407 Figure control for a segmented telescope mirror]. United States: N. p., 1979. Web. doi:10.2172/6194407. Some examples:

{{reflist}}

• Hexagonal number
• Magic hexagon
• Star number

{{Figurate numbers}}

{{Classes of natural numbers}}

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Category:Figurate numbers

Category:Integer sequences