Golfscript - 46

  / /__  4/{).+: ;.{  ++}%{.+}%+~ ]}@~(*n*

Golfscript - 47

  / /__  4/): ;{  +: ;.{  ++}%{.+}%+}@~(*n*

Golfscript - 48

   :  / /__\ +4/{2 *: ;.{  ++}%{.+}%+}@~(*n*

Golfscript - 51

~   :  / /__\ +4/(,{;2 *: ;.{  ++}%{.+}%+}%;n*

Same algorithm as my shorter python ( and ruby ) answer

Golfscript - 78

2~(?,{-1*}$1: ;{"  ":$*. 2base.{[$$+  /  ]=}%n+@@{[$$+"/__\"]=}%n .2*^: ;}%

Same algorithm as my longer python solution

This one has significant newlines

2~(?,{-1*}$1: ;{"  ":
*. 2base.{[
2*  /  ]=}%n+@@{[
2*"/__\"]=}%n .2*^: ;}%

J

46 characters, reading from stdin.

(,.~,~[,.~   $~#,#)^:(<:".1!:1]3)  / ,: /__ 

always delimits sentences, which made it impossible to fit inside S₃ (only 54 characters to play with). S₄ is a bit big at 162, so I padded it to fit. Serendipitously, / is a legal adverb. ☺

               /
              i=:3
             /  /
            %r=:1!:1
           /      /
          t=:]    [r+i
         /  /  /  /
        b=:  / ,: /__ 
       /              /
      i=:1            -".t
     /  /          /  /
    h=:(   $        ~#,#),.]
   /      /      /      /
  s=:(    h^:1    ,d=:    ,.~)
 /  /  /  /  /  /  /  /
(,,&(10{a.)"1[s^:(-i)b)(1!:2)(4)

Sorry I m late. This is based on A. Rex s Perl solution:

                           &I                               
                          ;for                              
                         $x  (2                             
                        ..<>){$E                            
                       .=      $E                           
                      ;my$    y;3*                          
                     33  +3  **  3;                         
                    s".+"$y.=$n.$&x2                        
                   ,$              E.                       
                  $&.$            E"ge                      
                 ;;  $_          .=  $y                     
                }print;;        sub I{($                    
               E,      $n      ,$      F,                   
              $B,$    U)=(    $",$    /,qw                  
             (/      _  ))  ;$  _=  $E  .$                 
            F.$B.$E.$n.$F.$U.$U.$B};33333333                

Go, 273 characters

package main
import(f"fmt";"os";s"strconv";)func main(){var
t=[2]string{" /\ ","/__\"};
n,_:=s.Atoi(os.Args[1]);a:=1;N:=a<<uint(n);for
N>0{N-=2;for
k:=0;k<2;k++{for
j:=0;j<N;j++{f.Print(" ")}b:=a;for
b>0{o:=t[k];if
b&1==0{o="    "}f.Print(o);b>>=1}f.Print("
")}a^=a*2}}

Whitespace is all significant.

Unminized with gofmt sierpinski-3.go | perl -p -e s/ / /g :

package main

import (
    "fmt";
    "os";
    "strconv";
)

func main() {
    var t = [2]string{" /\ ", "/__\"};
    n, _ := strconv.Atoi(os.Args[1]);
    a := 1;
    N := a << uint(n);
    for N > 0 {
        N -= 2;
        for k := 0; k < 2; k++ {
            for j := 0; j < N; j++ {
                fmt.Print(" ")
            }
            b := a;
            for b > 0 {
                o := t[k];
                if b&1 == 0 {
                    o = "    "
                }
                fmt.Print(o);
                b >>= 1;
            }
            fmt.Print("
");
        }
        a ^= a * 2;
    }
}

I got a good hint for Go golf here.

Python - 102

a=" / ","/__\"
j=   
for n in~-input()*j:j+=j;a=[j+x+j for x in a]+[x*2for x in a]
print"
".join(a)

Python - 105

a=" / ","/__\"
j=   
for n in(input()-1)*j:j+=j;a=[j+x+j for x in a]+[x+x for x in a]
print"
".join(a)

Python - 109

a=" / ","/__\"
for n in range(1,input()):j=   *2**n;a=[j+x+j for x in a]+[x+x for x in a]
print"
".join(a)

Python2.6 - 120

N=1<<input()
a=1
while N:
 N-=2
 for s in" / ","/__\":print   *N+bin(a)[2:].replace( 0 ,   *4).replace( 1 ,s)
 a=a^a*2

Perl, 82 strokes

This version no longer prints a trailing newline. Only the first newline is necessary:

$_=  / 
/__\ ;
for$x(2..<>){
my$y;
$".=$";
s#.+#$y.=$/.$&x2,$".$&.$"#ge;
$_.=$y
}
print

If command-line switches are allowed, then by traditional Perl golf scoring, this is 77+3 strokes (the first newline is literal):

#!perl -p
$=  / 
/__\ ;
$y="",
$".=$",
$=~s#.+#$y.=$/.$&x2,$".$&.$"#ge,
$.=$y
for 2..$_

Please feel free to edit my answer if you find an improvement.

Haskell, 153 149 137 125 118 112 characters:

Using tail recursion:

(%)=zipWith(++)
p="  ":p
g t _ 1=t
g t s(n+1)=g(s%t%s++t%t)(s%s)n
main=interact$unlines.g[" /\ ","/__\"]p.read

earlier version, @118 characters:

(%)=zipWith(++)
f 1=[" /\ ","/__\"]
f(n+1)=s%t%s++t%t where t=f n;s=replicate(2^n)   :s
main=interact$unlines.f.read

Using the (justly deprecated!) n+k pattern saved 4 characters.

I like how it comes out halfway readable even in compressed form.

edit:old main

main=do{n<-getLine;putStr$unlines$f$read n}

Perl

94 characters when newlines are removed.

$c=2**<>;$=$/;for$a(0..--$c){print$"x($c-$a&~1),
map$_*2&~$a?$"x4:$a&1? /__\ :  /  ,0..$a/2}

Ruby — 85

a=  /  , /__\ 
j=   
2.upto(gets.to_i){j+=j;a=a.map{|x|j+x+j}+a.map{|x|x+x}}
puts a

(a=2**gets.to_i).times{|y|puts" "*(a-y-1)+(0..y).map{|x|~y&x>0?    :y%2>0?x%2>0? _\ : /_ : /\ }*  }

Python, 135 chars

S=lambda n:[" /\ ","/__\"]if n==1 else[" "*(1<<n-1)+x+" "*(1<<n-1)for x in S(n-1)]+[x+x for x in S(n-1)]
for s in S(input()):print s

MATLAB - 64 characters (script version)

This assumes that you have the variable N already defined in your workspace:

A=[  /  ; /__ ];for i=1:N-1,B=32*ones(2^i);A=[B A B;A A];end;A

MATLAB - 78 characters (m-file function version)

Pass N as an argument to the function s:

function A=s(N),A=[  /  ; /__ ];for i=1:N-1,B=32*ones(2^i);A=[B A B;A A];end

C

Same algorithm as the Perl answer, but weighing in heavier, at 131 necessary characters.

a,b;main(c,v)char**v;{c=1<<atoi(v[1]);for(a=0;a<c;a++,puts(""))
for(b=c;b--;write(1,b&~a?"    ":a&1?"/__\":" /\ ",4-2*(b>a)))--b;}

I thought write(1,…) was UNIX API, but this seems to compile and run fine on Windows too.

If you replace char by int, it saves one character and still works, but it s of questionable legality.

Logo (not exactly following the requirements): 47 characters

to F:n if:n[repeat 3[F(:n-1)fd 2^:n rt 120]]end

I tested this only with http://www.calormen.com/Logo/ so I don t know if it s portable. It doesn t follow the requirements, but surely logo must be the appropriate language here? :) I love that at the time of writing logo is one character short of equalling golfscript and J.

Lua, 139 characters

t={" /\ ","/__\"}for i=2,(...)do for j=1,#t do
t[#t+1]=t[j]:rep(2)k=(" "):rep(#t[j]/2)t[j]=k..t[j]..k end end
print(table.concat(t,"
"))

Nroff, 542

$ nroff -rn=5 file.n

.pl 1
.nf
.de b
. nr i 0
. while d\$1\ni {
.   \$3 \$1\ni \$2\ni
.   nr i +1
. }
..
.de push
. nr i 0
. while d\$2\ni {
.   nr i +1
. }
. nr j 0
. while d\$1\nj {
.   ds \$2\ni &\*[\$1\nj]
.   nr i +1
.   nr j +1
. }
..
.ds l0 & /[rs] &
.ds l1 "/__[rs]
.ds s & 
.de o
. ds \$2 &\*s\*[\$1]\*s
..
.de p
. ds \$2 &\*[\$1]\*[\$1]
..
.de assign
. ds \$2 &\*[\$1]
..
.nr a 2
.while 
a<=
n {
. ds s &*s*s
. b l m o
. b l n p
. b m l assign
. push n l
. nr a +1
.}
.de t
\*[\$1]
..
.b l zz t

F#, 225 chars

let rec p n=if n=1 then" "else p(n-1)+p(n-1)
and S n=if n=1 then[" /\ ";"/__\"]else let s=S(n-1)in List.append(List.map(fun s->p(n)+s+p(n))s)(List.map(fun x->x+x)s)
for s in S(int(System.Console.ReadLine()))do printfn"%s"s

Clojure: 174 characters

Algorithm stolen from others above.

(doseq[q((fn f[n](if(= n 1)[" /\ ""/__\"](let[z(f(dec n))](concat(map #(let[y(repeat(Math/pow 2(dec n)) )](apply str(concat y % y)))z)(map str z z)))))(read))](println q))

38 of those characters are parentheses. : (

(doseq [q ((fn f [n]
           (if (= n 1)
             [" /\ " "/__\"]
             (let [z (f (dec n))]
               (concat
                (map #(let [y (repeat (Math/pow 2 (dec n)) )]
                        (apply str (concat y % y))) z)
                (map str z z))))) (read))] 
  (println q))

Python, 120 characters (recursive solution)

S=lambda n:n<2and[" / ","/__\"]or[" "*n+x+" "*n for x in S(n/2)]+[x+x for x in S(n/2)]
print"
".join(S(1<<input()-1))

I started putting on the green where @fserb left off...

GolfScript (45 44 chars)

~(  / /__  4/)@{.+.{[2$.]*}%{.+}%+}*;n*

Similar to gnibbler s solution. My initial attempt was already quite similar, and then I looked at his and borrowed some ideas.

Python, 186 chars (UNIX line termination)

for j in range(1,n):
 for s in p:
  print s
 x=2**j;y=2*x;p.extend([  ]*x)
 for i in range(y-1,-1,-1):
  if i<x:
   s=   *x;p[i]=s+p[i]+s
  else:
   q=p[i-x];p[i]=q+q

Prolog, 811 Chars

:- module(sierpinsky, [draw/1]).

% draw(+Level)
draw(N) :- K is 2^(N+1)-1,
  for(Line, 0, K),
  draw2(N, Line, true, nl),
  fail.
draw(_).

% draw2(+Level, +Line, +Before, +After)
draw2(0, 0, Before, After) :- !,
  Before, write(  /\  ), After.
draw2(0, 1, Before, After) :- !,
  Before, write( /__\ ), After.
draw2(N, Line, Before, After) :- N>0, K is 2^N, Line < K, !, M is N-1,
  draw2(M, Line, (Before, tab(K)), (tab(K), After)).
draw2(N, Line, Before, After) :- N>0, K is 2^N, Line >= K, !, M is N-1,
  Line2 is Line - K,
  draw2(M, Line2, Before, draw2(M, Line2, true, After)).

% for(+Variable, +Integer, +Integer)
for(V, N, M) :- N =< M, V = N.
for(V, N, M) :- N < M, K is N+1, for(V, K, M).

% tab(+Integer)
tab(N) :- for(_, 1, N), write(   ), fail.
tab(_).