A centered decagonal number is a
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
:
Thus, the first few centered decagonal numbers are
:
1,
11,
31,
61,
101 101 may refer to:
* 101 (number), the number
* AD 101, a year in the 2nd century AD
* 101 BC, a year in the 2nd century BC
It may also refer to:
Entertainment
* ''101'' (album), a live album and documentary by Depeche Mode
* "101" (song), a ...
,
151, 211, 281, 361, 451, 551, 661, 781,
911, 1051, ...
Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n'' − 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 1.
Another consequence of this relation to triangular numbers is the simple
recurrence relation for centered decagonal numbers:
:
where
:
Generating Function
The generating function of the centered decagonal number is
Continued fraction forms
has the
continued fraction expansion
n-3;
See also
*
rdinary decagonal number
References
{{Classes of natural numbers
Figurate numbers