05AB1E, 21 bytes

3*s≠-L·<sÅ0«Âø€à'*×.C

Try it online!
or as a Test Suite

Explanation

3* # multiply input by 3
 s≠- # subtract 1 if the input isn't 1
 L # push range [1 ... (3*n-(n!=1))]
 ·< # multiply by 2 and subtract 1 to get odd numbers
 sÅ0« # append n zeroes
 Âø # zip with a reversed copy
 ۈ # get the largest number in each pair
 '*× # repeat "*" for each number in the list
 .C # format centered

Haskell, 114 bytes

Builds a function g which takes an number and produces a IO monad that prints the star to STDOUT. I think this is ok.

f=replicate
a%b=mapM_(n->putStrLn$f(a-n)' '++f(2*n-3)'*')$zipWith max<*>reverse$[2..a]++f b 0
g 1=4%1
g a=(3*a)%a

Try it online!

Explanation

First lets talk about the lambda.

n->putStrLn$f(a-n)' '++f(2*n-1)'*'

This takes a number, n, to be drawn as stars. We print twice that many stars and then 1 more and pad it on the right to the size of the image. We pad this on the right by a spaces to center the line of stars. We can use this lambda to draw each line.

From this lambda we create (%). (%) starts with doing mapM_ with our lambda to turn a profile into the shape.

Now all we need to do is make a list of the profile for the star. We can do this by making a triangle first with [1..a], then padding it with some zeros ++replicate b 0. If we take the profile of the triangle and reverse it we get the other half of the star. To super impose them we just make a new profile where each entry is the max of the two triangles. This is zipWith max.

We then call this in one of two ways: as 3%1 for input of 1 and with (3*a-1)%a otherwise.

From here we do a little bit of fiddling with some of the values to shave some bytes. Since 3*a-1 is rather long we offset some of our other values by 1 so that everything cancels and we get the intended behavior with 3*a instead. Namely we start our list at 2 instead of 1 and do 2*n-3 instead of 2*n-1 to make up for the change.

Alternative version, 114 bytes

This one builds a point-free function (%)<*>min 2

f=replicate
a%b=mapM_(n->putStrLn$f(3*a-n)' '++f(2*(n-b)+1)'*')$zipWith max<*>reverse$[b..3*a]++f a 0
(%)<*>min 2

Try it online!

Handles all $N>1$, 98 bytes

f=replicate
g a=mapM_(n->putStrLn$f(3*a-n)' '++f(2*n-3)'*')$zipWith max<*>reverse$[2..3*a]++f a 0

Try it online!

Handles all cases like the $N=1$ case, 98 bytes

f=replicate
g a=mapM_(n->putStrLn$f(3*a-n)' '++f(2*n-1)'*')$zipWith max<*>reverse$[1..3*a]++f a 0

Try it online!

R, 125 117 107 bytes

function(n,S=3*n+!n-1,P=pmax(I<-c(2:S*2-3,!1:n),rev(I)),`~`=strrep)cat(paste0(' '~S-2-P%/%2,'*'~P),sep='
')

Try it online!

• -18 thanks to @Giuseppe

Previous (different) approach :

R, 150 148 136 135 130 128 bytes

function(n,G=n<2,K=4*n-1+G)for(i in 1:K-1)cat(`[<-`(rep(' ',W<-3*n-2+G),1+W+c(-i:i*(i<K-n),-(j=K-i-1):j*(i>=n)),'*'),sep='','
')

Try it online!

• -14 thanks to @Kirill L.


• -1 thanks to @t-clausen.dk


• -7 thanks to @Giuseppe

JavaScript (ES8), 101 bytes

n=>(k=3*n+!~-n,g=y=>++y<k+n?`
`.padEnd(w=k-Math.max(y>n&&n-y+k,y<k&&y)).padEnd(2*k+~w,'*')+g(y):'')``

Try it online!

Python 2, 101 99 97 bytes

n=input()
x=2*(n>1)
for i in range(1,8*n,2):print('*'*[i,8*n-i-x][i+x>n*6or i/n/2%2]).center(6*n)

Try it online!

_{-2 bytes, thanks to Lynn}

Jelly, 21 bytes

×3’+ỊR;Ṭ»Ṛ$”*ẋz⁶ṚZŒBY

A full program accepting a positive integer which prints to STDOUT.

Try it online! Or see a test-suite.

How?

×3’+ỊR;Ṭ»Ṛ$”*ẋz⁶ṚZŒBY - Main Link: integer, n e.g. 3
 3 - three 3
× - multiply (n by) 9
 ’ - decrement 8
 Ị - insignificant (abs(n)<=1)? 0
 + - add 8
 R - range [1,2,3,4,5,6,7,8]
 Ṭ - un-truth (n) [0,0,1]
 ; - concatenate [1,2,3,4,5,6,7,8,0,0,1]
 $ - last two links as a monad:
 Ṛ - reverse [1,0,0,8,7,6,5,4,3,2,1]
 » - maximum (vectorises) [1,2,3,8,7,6,7,8,3,2,1]
 ”* - an asterisk character '*' 
 ẋ - repeat (vectorises) ["*","**",...]
 ⁶ - a space character ' '
 z - transpose with filler ["***********"," ********* ",...]
 Ṛ - reverse [" * * "," ** ** ",...]
 Z - transpose [" *"," **",...]
 ŒB - bounce (vectorises) [" * "," *** ",...]
 Y - join with newline characters " * n *** n..."
 - implicit print

Canvas, 25 23 bytes

Ｒ：｛⁸３×４Ｍ∔］∔｛*×］↔⁸１≡？╪］┼

Try it here!

15 bytes without handling 1

Jelly, 21 bytes

×’»ɗ3”*xⱮz⁶ɓ⁶x;»Ṛ$ŒBY

Try it online!

A monadic link accepting a single integer as its left argument and returning a newline-separated Jelly string with the star as its output. When run as a full program implicitly prints the star to STDOUT.

Explanation

 ɗ3 | Last three links as a dyad with 3 as right argument:
× | Multiply (by 3)
 ’ | Decrease by 1
 » | Maximum of this (and 3)
 ”*xⱮ | An implicit range from 1 to this many asterisks
 z⁶ | Zip with space as filler
 ɓ | Start a new dyadic chain with the input as left argument and the list of asterisks as right argument
 ⁶x | Input many spaces
 ; | Concatenated to the asterisk list
 $ | Last two links as a monad:
 »Ṛ | Maximum of this list and its reverse
 ŒB | Bounce each list (i.e. mirror it without duplicating the middle entry)
 Y | Join with newlines

Charcoal, 25 bytes

Ｇ↙θ←⁺⊗θ¬⊖θ↑⊗θ↘⊕θ*‖Ｏ¬Ｃ⁰¬⊖θ

Try it online! Link is to verbose version of code. Explanation:

Ｇ↙θ←⁺⊗θ¬⊖θ↑⊗θ↘⊕θ*

Draw an irregular pentagon representing the top right quarter of the star, but special-casing 1 to make the row an extra column wider.

‖Ｏ¬

Reflect to complete the star.

Ｃ⁰¬⊖θ

More special-casing to make the star for 1 an extra row taller.

Alternative solution, also 25 bytes:

∧¬⊖θ*Ｇ↗↓⊖׳Ｎ*‖Ｏ‖ＯＯ↓∧⊖θ⊖⊗θ

Try it online! Link is to verbose version of code. Explanation:

∧¬⊖θ*

Print an extra * for the case of 1.

Ｇ↗↓⊖׳Ｎ*

Draw the left half of a triangle of the appropriate size.

‖Ｏ

Reflect to complete the triangle.

‖ＯＯ↓∧⊖θ⊖⊗θ

Overlap it with its reflection, except in the case of 1, in which case just reflect it.

14 bytes without special-casing for 1:

Ｇ<⊖׳Ｎ*‖ＯＯ↑⊖⊗θ

Try it online! Link is to verbose version of code. Explanation:

Ｇ<⊖׳Ｎ*

Draw a triangle of the appropriate size.

‖ＯＯ↑⊖⊗θ

Overlap it with its reflection.

Perl 6, 74 bytes

[R,] $_o.&

Try it online!

Literally creates a triangle with the right proportions and overlaps it with a upside-down copy using the string or operator (~|). Outputs as a list of lines with a leading and trailing line whitespace.

Explanation:

 {.& # Anonymous code block
 ( )xx$_*3-($_>1) # Repeat n*3-(n!=1) times
 ' 'x--$+$_*3 # With a decreasing indentation
 ~'*'x$++*2+1 # Append an increasing triangle
 # This creates the triangle
 ,|($ xx$_) # And add some padding lines
 o # Pass the triangle to the combining function
 Z~| # Zip string bitwise or
 $_ # The list
 [R,] $_ # With its reverse

J, 53 50 bytes

' *'~[:(+..)"1*@<:>:/~@i.@-~3*]

Try it online!

ungolfed

' *' ~ [: (+. .)"1 *@<: >:/~@i.@-~ 3 * ]

how

Use a function table (like a 3rd grade times table) to construct half the triangle by using >: (greater than or equal) as the function. Then reverse each row, chop of the last column, and stitch the two sides together to get the full triangle (but made of 1 and 0). Add n rows of zeros at the bottom. Finally reverse the whole thing, and overlay it on the original, using boolean or +. to get the result. Then turn the 1 to * and 0 to spaces.

T-SQL, 194 bytes

@ is the input value

@c handles the width of the top triangle

@d handles the width bottom triangle

@e contains the output either @c or @d - this saves a few bytes

@f handles the special case of 1 as input. @c*@=3 determines when to use @f. 5 bytes cheaper than writing @c=3and @=1

DECLARE @ INT=8

,@c INT=1,@e INT=1,@d INT,@f INT=0SET @d=@*8-3r:PRINT
space(@*3-@e/2+@f/2)+replicate('*',@e-@f)SELECT
@c=nullif(@c,@*6-3)+2,@f=iif(@c*@=3,2,0),@d-=2-@f,@e=iif(@c>@d
or @c/2<@,@c,@d)IF @d>0goto r

Try it online

Japt -R, 25 bytes

+5 bytes for n=1 :

õ cUon3*U-´UÎ)®ç* êÃê!U û

Try it

õ cUon3*U-´UÎ)®ç* êÃê!U û :Implicit input of integer U
õ :Range [1,U]
 c :Concatenate
 Uo : Range [0,U)
 n : Subtract each from
 3*U- : Multiply U by 3 and subtract
 ´U : Decrement U
 Î : Get sign
 ) :End concat
 ® :Map each Z
 ç* : Repeat "*" Z times
 ê : Palindromise
 Ã :End map
 ê!U :If decremented U is 0, append reverse, else, palindromise
 û :Centre pad each line with spaces to the length of the longest
 :Implicitly join with newlines and output