Editing Centered decagonal number (section)

{{Short description|Centered figurate number that represents a decagon with a dot in the center}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}
[[Image:Centered decagonal number.svg|280px|right]]

A '''centered decagonal number''' is a [[centered number|centered]] [[figurate number]] that represents a [[decagon]] with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula

:<math>5n^2-5n+1 \, </math>

Thus, the first few centered decagonal numbers are

:[[1 (number)|1]], [[11 (number)|11]], [[31 (number)|31]], [[61 (number)|61]], [[101 (number)|101]], [[151 (number)|151]], 211, 281, 361, 451, 551, 661, 781, [[911 (number)|911]], 1051, ... {{OEIS|id=A062786}}

Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n''&nbsp;&minus;&nbsp;1)th [[triangular number]] by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in&nbsp;1.

Another consequence of this relation to triangular numbers is the simple [[recurrence relation]] for centered decagonal numbers:
:<math>CD_{n} = CD_{n-1}+10n ,</math>
where
:<math>CD_0 = 1 .</math>