Editing Polygonal number (section)

==Formula==
[[File:visual_proof_polygonal_numbers.svg|thumb|An ''s''-gonal number greater than 1 can be decomposed into ''s''&minus;2 triangular numbers and a natural number.]]If {{mvar|s}} is the number of sides in a polygon, the formula for the {{mvar|n}}th {{mvar|s}}-gonal number {{math|''P''(''s'',''n'')}} is

:<math>P(s,n) = \frac{(s-2)n^2-(s-4)n}{2}</math>

The {{mvar|n}}th {{mvar|s}}-gonal number is also related to the triangular numbers {{math|''T''<sub>''n''</sub>}} as follows:<ref name=":0">{{Cite book |last1=Conway |first1=John H. |title=The Book of Numbers |last2=Guy |first2=Richard |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4612-4072-3 |pages=38–41 |language=en |author-link=John Horton Conway |author-link2=Richard K. Guy}}</ref>

:<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>

Thus:

:<math>\begin{align}
P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\
P(s+1,n) - P(s,n) &= T_{n-1} = \frac{n(n-1)}{2}\, ,\\
P(s+k,n) - P(s,n) &= k T_{n-1} = k\frac{n(n-1)}{2}\, .
\end{align}</math>

For a given {{mvar|s}}-gonal number {{math|''P''(''s'',''n'') {{=}} ''x''}}, one can find {{mvar|n}} by

:<math>n = \frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)}</math>

and one can find {{mvar|s}} by

:<math>s = 2+\frac{2}{n}\cdot\frac{x-n}{n-1}</math>.

===Every hexagonal number is also a triangular number===
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|Description = [[Proof without words]] that hexagonal numbers are odd-sided triangular numbers
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Applying the formula above:
:<math>P(s,n) = (s-2)T_{n-1} + n </math>

to the case of 6 sides gives:
:<math>P(6,n) = 4T_{n-1} + n </math>

but since:
:<math>T_{n-1} = \frac{n(n-1)}{2} </math>

it follows that:
:<math>P(6,n) = \frac{4n(n-1)}{2} + n = \frac{2n(2n-1)}{2} = T_{2n-1}</math>

This shows that the {{mvar|n}}th hexagonal number {{math|''P''(6,''n'')}} is also the {{math|(2''n'' − 1)}}th triangular number {{math|''T''<sub>2''n''−1</sub>}}.  We can find every hexagonal number by simply taking the odd-numbered triangular numbers:<ref name=":0" />

:'''1''', 3, '''6''', 10, '''15''', 21, '''28''', 36, '''45''', 55, '''66''', ...