Acentered decagonal number is acenteredfigurate number that represents adecagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number forn is given by the formula

${\displaystyle 5n^2-5n+1}$

Thus, the first few centered decagonal numbers are

1,11,31,61,101,151, 211, 281, 361, 451, 551, 661, 781,911, 1051, ... (sequenceA062786 in theOEIS)

Like any other centeredk-gonal number, thenth centered decagonal number can be reckoned by multiplying the (n − 1)thtriangular number byk, 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 1.

Another consequence of this relation to triangular numbers is the simplerecurrence relation for centered decagonal numbers:

${\displaystyle C{D}_n=C{D}_{n-1}+10n,}$

where

${\displaystyle C{D}_0=1.}$

• N is a Centered decagonal number iff 20N + 5 is aSquare number.

The generating function of the centered decagonal number is${\displaystyle \frac{x\ast (1+8x+x^2)}{(1-x{)}^3}}$

${\displaystyle \sqrt{5C{D}_n}}$ has thesimple continued fraction [5n-3;{2,2n-2,2,10n-6}].

• [ordinary]decagonal number

Deza, Elena; Deza, Michel Marie (November 20, 2011). "1.6".Figurate Numbers. WORLD SCIENTIFIC.doi:10.1142/8188.ISBN 978-981-4355-48-3.

• Figurate numbers

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