Centered hexagonal number

<h2 id="formula">Formula</h2>

<p>The nth centered hexagonal number is given by the formula<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

H
        (
        n
        )
        =
        
          n
          
            3
          
        
        −
        (
        n
        −
        1
        
          )
          
            3
          
        
        =
        3
        n
        (
        n
        −
        1
        )
        +
        1
        =
        3
        
          n
          
            2
          
        
        −
        3
        n
        +
        1.
        
      
    
    {\displaystyle H(n)=n^{3}-(n-1)^{3}=3n(n-1)+1=3n^{2}-3n+1.\,}

<p>Expressing the formula as
</p>

H
        (
        n
        )
        =
        1
        +
        6
        
          (
          
            
              
                n
                (
                n
                −
                1
                )
              
              2
            
          
          )
        
      
    
    {\displaystyle H(n)=1+6\left({\frac {n(n-1)}{2}}\right)}

<p>shows that the centered hexagonal number for n is 1 more than 6 times the (<i>n</i> − 1)th <a href="/facts/Triangular_number/KHonHtQJ">triangular number</a>.
</p><p>In the opposite direction, the <i>index</i> n corresponding to the centered hexagonal number 
  
    
      
        H
        =
        H
        (
        n
        )
      
    
    {\displaystyle H=H(n)}
  
 can be calculated using the formula
</p>

n
        =
        
          
            
              3
              +
              
                
                  12
                  H
                  −
                  3
                
              
            
            6
          
        
        .
      
    
    {\displaystyle n={\frac {3+{\sqrt {12H-3}}}{6}}.}

<p>This can be used as a test for whether a number H is centered hexagonal: it will be if and only if the above expression is an integer.
</p>
<h2 id="recurrence-and-generating-function">Recurrence and generating function</h2>
<p>The centered hexagonal numbers 
  
    
      
        H
        (
        n
        )
      
    
    {\displaystyle H(n)}
  
 satisfy the <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

H
        (
        n
        +
        1
        )
        =
        H
        (
        n
        )
        +
        6
        n
        .
      
    
    {\displaystyle H(n+1)=H(n)+6n.}

<p>From this we can calculate the <a href="/facts/Generating_function/K7UVjnjL">generating function</a> 
  
    
      
        F
        (
        x
        )
        =
        
          ∑
          
            n
            ≥
            0
          
        
        H
        (
        n
        )
        
          x
          
            n
          
        
      
    
    {\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}}
  
. The generating function satisfies
</p>

F
        (
        x
        )
        =
        x
        +
        x
        F
        (
        x
        )
        +
        
          ∑
          
            n
            ≥
            2
          
        
        6
        n
        
          x
          
            n
          
        
        .
      
    
    {\displaystyle F(x)=x+xF(x)+\sum _{n\geq 2}6nx^{n}.}

<p>The latter term is the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> of 
  
    
      
        
          
            
              6
              x
            
            
              (
              1
              −
              x
              
                )
                
                  2
                
              
            
          
        
        −
        6
        x
      
    
    {\displaystyle {\frac {6x}{(1-x)^{2}}}-6x}
  
, so we get
</p>

(
        1
        −
        x
        )
        F
        (
        x
        )
        =
        x
        +
        
          
            
              6
              x
            
            
              (
              1
              −
              x
              
                )
                
                  2
                
              
            
          
        
        −
        6
        x
        =
        
          
            
              x
              +
              4
              
                x
                
                  2
                
              
              +
              
                x
                
                  3
                
              
            
            
              (
              1
              −
              x
              
                )
                
                  2
                
              
            
          
        
      
    
    {\displaystyle (1-x)F(x)=x+{\frac {6x}{(1-x)^{2}}}-6x={\frac {x+4x^{2}+x^{3}}{(1-x)^{2}}}}

F
        (
        x
        )
        =
        
          
            
              x
              +
              4
              
                x
                
                  2
                
              
              +
              
                x
                
                  3
                
              
            
            
              (
              1
              −
              x
              
                )
                
                  3
                
              
            
          
        
        .
      
    
    {\displaystyle F(x)={\frac {x+4x^{2}+x^{3}}{(1-x)^{3}}}.}

<h2 id="properties">Properties</h2>

<p>In <a href="/facts/Base_10/7omfVNIM">base 10</a> one can notice that the hexagonal numbers' rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with <a href="/facts/Periodic_sequence/eIJhDfJn">period</a> 5).
This follows from the <a href="/facts/Triangular_number/KHonHtQJ">last digit of the triangle numbers</a> (sequence A008954 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>) which repeat 0-1-3-1-0 when taken modulo 5.
In <a href="/facts/Base_6/BpNMv2Lp">base 6</a> 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.
</p><p>The sum of the first n centered hexagonal numbers is <a href="/facts/Cube_(algebra)/SqmlEEog"><i>n</i>3</a>. That is, centered hexagonal <a href="/facts/Pyramidal_number/tcRjsACG">pyramidal numbers</a> and <a href="/facts/Cubic_number/SqmlEEog">cubes</a> 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 <a href="/facts/Figurate_number/f1gdGcja">gnomon</a> of the cubes. (This can be seen geometrically from the diagram.)  In particular, <a href="/facts/Prime_number/L20FLkgn">prime</a> centered hexagonal numbers are <a href="/facts/Cuban_prime/CBxry27b">cuban primes</a>.
</p><p>The difference between (2<i>n</i>)2 and the nth centered hexagonal number is a number of the form 3<i>n</i>2 + 3<i>n</i> − 1, while the difference between (2<i>n</i> − 1)2 and the nth centered hexagonal number is a <a href="/facts/Pronic_number/klUXKkSd">pronic number</a>.
</p>
<h2 id="applications">Applications</h2>

<p>Many <a href="/facts/Segmented_mirror/UTGzsy5u">segmented mirror</a> <a href="/facts/Reflecting_telescope/5Y4aXFvF">reflecting telescopes</a> 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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> Some examples:
</p>
<table><tbody><tr><th>Telescope</th><th>Number ofsegments</th><th>Numbermissing</th><th>Total</th><th><i>n</i>-th centeredhexagonal number</th></tr><tr><td><a href="/facts/Giant_Magellan_Telescope/A1g7gz9P">Giant Magellan Telescope</a></td><td>7</td><td>0</td><td>7</td><td>2</td></tr><tr><td><a href="/facts/James_Webb_Space_Telescope/w4GAWnWX">James Webb Space Telescope</a></td><td>18</td><td>1</td><td>19</td><td>3</td></tr><tr><td><a href="/facts/Gran_Telescopio_Canarias/Kc8D0DEh">Gran Telescopio Canarias</a></td><td>36</td><td>1</td><td>37</td><td>4</td></tr><tr><td><a href="/facts/Guido_Horn_d%2527Arturo/4rvCeJte">Guido Horn d'Arturo</a>'s prototype</td><td>61</td><td>0</td><td>61</td><td>5</td></tr><tr><td><a href="/facts/Southern_African_Large_Telescope/5QJetKKA">Southern African Large Telescope</a></td><td>91</td><td>0</td><td>91</td><td>6</td></tr></tbody></table>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hexagonal_number/7yXaNygK">Hexagonal number</a></li>
<li><a href="/facts/Magic_hexagon/n4Nd1Kla">Magic hexagon</a></li>
<li><a href="/facts/Star_number/ETasl301">Star number</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". J. Rec. Math. 16: 191–193. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Deza, Elena; Deza, M. (2012). Figurate Numbers. World Scientific. pp. 47–55. ISBN 978-981-4355-48-3. <a href="978-981-4355-48-3" target="_blank">978-981-4355-48-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Deza, Elena; Deza, M. (2012). Figurate Numbers. World Scientific. pp. 47–55. ISBN 978-981-4355-48-3. <a href="978-981-4355-48-3" target="_blank">978-981-4355-48-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Deza, Elena; Deza, M. (2012). Figurate Numbers. World Scientific. pp. 47–55. ISBN 978-981-4355-48-3. <a href="978-981-4355-48-3" target="_blank">978-981-4355-48-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Mast, T. S. and Nelson, J. E. Figure control for a segmented telescope mirror. United States: N. p., 1979. Web. doi:10.2172/6194407. <a href="https://osti.gov/servlets/purl/6194407" target="_blank">https://osti.gov/servlets/purl/6194407</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Centered hexagonal number open-in-new

Centered hexagonal number