YASEPL, 51 bytes

=d$=n=i'=j$`1=e$d=z$n%d*!-z+n!n$e!j+}4,i#n#divide>d

explanation:

=d$=n=i'=j$`1=e$d=z$n%d*!-z+n!n$e!j+}4,i#n#divide>d   packed
=d$                                                   denominator = d = 1
   =n                                                 numerator = n = 0
     =i'                                              get input, set to i
        =j$`1                       }4,i              j = 1, while j < i...
             =e$d                                       e = d
                 =z$n%d*                                z = (n%d)*2
                        !-z+n                           d = d - z + n
                             !n$e                       n = e
                                 !j+                    increment j
                                        #n#divide>d   output fraction
       

APL(NARS), 49 chars

r←f w;k
r←1x⋄→0×⍳w≤1⋄r←1+k←f⌊w÷2⋄→0×⍳2∣w⋄r←÷1+÷k

It seems r(x)=(1 if x=1;1+r((x-1)/2) if x is odd;1/(1+1/r(x/2)) if x is even)

"⌊w÷2" is ok as "w÷2" if w is even, or "(w-1)÷2" if w is odd.

1x or 1 with type rational is the base of induction, and from this value are built all other values that maintain the same type of base of induction.

//8+41=49

  f ¨⍳10
1 1r2 2 1r3 3r2 2r3 3 1r4 4r3 3r5 
  f 100
7r19 
  f 1000
11r39 

TI-BASIC (TI-83 Plus), 69 bytes

1→D
Input I
For(J,1,I
D→E
N+D-2(N-Dint(N/D→D
E→N
End
2+int(log(N
Output(1,1,N
Output(1,Ans,"/
Output(1,Ans+1,D

assuming you clear your memory after every run of the program. outputs in the top left

Desmos, 212 bytes

^{Try it online! (link 1) Try it online! (link 2)} (just please note that due to Desmos' max recursion limit, you'll have to wait quite some time just to get the function to even load values properly)

s\left(n\right)=\left\{n=1:1,\operatorname{mod}\left(n,2\right)=0:s\left(\frac{n}{2}\right),s\left(\frac{n-1}{2}\right)+s\left(\frac{n+1}{2}\right)\right\}
r\left(n\right)=\frac{s\left(n\right)}{s\left(n+1\right)}

Readable version:

s(n)={n=1:1,mod(n,2)=0:s(n/2),s((n-1)/2)+s((n+1)/2)}
r(n)=s(n)/s(n+1)

How does it work? Well, we can define recursive functions in Desmos:$$s(n)=\{n=1:1,\operatorname{mod}(n,2)=0:s(n/2),s((n-1)/2)+s((n+1)/2)\}$$and now all we have to do is create \$r(n)=\frac{s(n)}{s(n+1)}\$, and we're done!

PARI/GP, 33 bytes

f(n)=if(n,(1+f(n\2)^q=n%2*2-1)^q)

Attempt This Online!

APL (Dyalog Unicode), 56 bytes (SBCS)

{1(⍵{s←⍺⊃⍵⋄t←⍵[n←⍺+1]⋄⍺=⍺⍺:,/⍕¨s'/'t⋄n∇⍵,(s+t),t})1 1}

Try it online!

Inner operator:

{                 ⍝ ⍺ is counter, ⍺⍺ is input, ⍵ is sequence
 s ← ⍺ ⊃ ⍵       ⍝ s is the ⍺th element of ⍵ (1-indexed), i.e. s(n)
⋄t ← ⍵[n←⍺+1]     ⍝ t is s(n+1). Also create variable n that holds ⍺+1
⋄⍺ = ⍺⍺:          ⍝ If counter equals the n we want to reach
  s ÷ t           ⍝ Return s(n) / s(n+1)
⋄n ∇ ⍵,(s+t),t   ⍝ Otherwise, call itself again with counter incremented
                 ⍝ and the new terms added to the sequence
                 ⍝ ∇ is self reference and , is concatenate
}

The outer function just calls the inner one with 1 as the initial value of the counter, 1 1 as the initial sequence, and its right argument as the left operand of the inner operator (n).

Maple

0: do 1/(1+floor(%)-frac(%))od;

Uses % which is the ditto operator reevaluating the last expression.

R, 84 bytes

function(n,K=c(1,1)){for(i in 1:n)K=c(K,K[i]+K[i+1],K[i+1])
paste0(K[i],"/",K[i+1])}

Try it online!

The older R implementation doesn't follow the specs, returning a floating point rather than a string, so here's one that does.

Mathematica, 108 106 101 bytes

(s={1,1};n=1;a=AppendTo;t=ToString;Do[a[s,s[[n]]+s[[++n]]];a[s,s[[n]]],#];t@s[[n-1]]<>"/"<>t@s[[n]])&

Jelly, 14 bytes

3ẋḶŒpḄċ
’Ç”/⁸Ç

Try it online!

Ooh, looks like I can beat the accepted answer by @Dennis, and in the same language at that. This works using a formula from OEIS: the number of ways to express (the number minus 1) in hyperbinary (i.e. binary with 0, 1, 2 as digits). Unlike most Jelly programs (which work either as a full program or a function), this one works only as a full program (because it sends part of the output to stdout, and returns the rest; when used as a full program the return value is sent to stdout implicitly, so all the output is in the same place, but this wouldn't work for a function submission).

This version of the program is very inefficient. You can create a much faster program that works for all input up to 2ⁿ via placing n just after the ẋ on the first line; the program has O(n × 3ⁿ) performance, so keeping n small is fairly important here. The program as written sets n equal to the input, which is clearly large enough, but also clearly too large in almost all cases (but hey, it saves bytes).

Explanation

As usual in my Jelly explanations, text in braces (e.g. {the input}) shows something that automatically filled in by Jelly due to missing operands in the original program.

Helper function (calculates the nth denominator, i.e. the n+1th numerator):

3ẋḶŒpḄċ
3ẋ       3, repeated a number of times equal to {the argument}
  Ḷ      Map 3 to [0, 1, 2]
   Œp    Take the cartesian product of that list
     Ḅ   Convert binary (or in this case hyperbinary) to a number
      ċ  Count number of occurrences of {the argument} in the resulting list

The first five bytes are basically just generating all possible hyperbinary numbers up to a given length, e.g. with input 3, the output of Ḷ is [[0,1,2],[0,1,2],[0,1,2]] so the cartesian product is [[0,0,0],[0,0,1],[0,0,2],[0,1,0],…,[2,2,1],[2,2,2]]. Ḅ basically just multiplies the last entry by 1, the penultimate entry by 2, the antepenultimate entry by 4, etc., and then adds; although this is normally used to convert binary to decimal, it can handle the digit 2 just fine, and thus works for hyperbinary too. Then we count the number of times the input appears in the resulting list, in order to get an appropriate entry in the sequence. (Luckily, the numerator and denominator follow the same sequence).

Main program (asks for the numerator and denominator, and formats the output):

’Ç”/⁸Ç
’Ç      Helper function (Ç), run on {the input} minus 1 (‘)
  ”/    {Output that to stdout}; start again with the constant '/'
    ⁸Ç  {Output that to stdout}; then run the helper function (Ç) on the input (⁸)

Somehow I keep writing programs that take almost as many bytes to handle I/O as they do to solve the actual problem…

Haskell, 78 77 65 58 bytes

Shamelessly stealing the optimized approach gives us:

(s#t)0=show s++'/':show t
(s#t)n=t#(s+t-2*mod s t)$n-1
1#1

Thanks to @nimi for golfing a few bytes using an infix function!

(Still) uses 0-based indexing.

The old approach:

s=(!!)(1:1:f 0)
f n=s n+s(n+1):s(n+1):(f$n+1)
r n=show(s n)++'/':(show.s$n+1)

Damn the output format... And indexing operators. EDIT: And precedence.

Fun fact: if heterogenous lists were a thing, the last line could be:

r n=show>>=[s!!n,'/',s!!(n+1)]

J, 29 bytes

([,'/',])&":&([:+/2|]!&i.-)>:

Usage

Large values of n require a suffix of x which denotes the use of extended integers.

   f =: ([,'/',])&":&([:+/2|]!&i.-)>:
   f 1
1/1
   f 10
3/5
   f 50
7/12
   f 100x
7/19
   f 1000x
11/39

Actually, 18 bytes

11,`│;a%τ@-+`nk'/j

Try it online!

This solution uses Peter's formula and is likewise 0-indexed. Thanks to Leaky Nun for a byte.

Explanation:

11,`│;a%τ@-+`nk'/j
11                  push 1, 1
  ,`│;a%τ@-+`n      do the following n times (where n is the input):
                      stack: [t s]
    │                 duplicate the entire stack ([t s t s])
     ;                dupe t ([t t s t s])
      a               invert the stack ([s t s t t])
       %              s % t ([s%t s t t])
        τ             multiply by 2 ([2*(s%t) s t t])
         @-           subtract from s ([s-2*(s%t) s t])
           +          add to t ([t+s-2*(s%t) t])
                      in effect, this is s,t = t,s+t-2*(s%t)
              k'/j  push as a list, join with "/"

C (GCC), 79 bytes

Uses 0-based indexing.

s(n){return!n?:n%2?s(n/2):s(-~n/2)+s(~-n/2);}r(n){printf("%d/%d",s(n),s(n+1));}

Ideone

Ruby, 49 bytes

This is 0-indexed and uses Peter Taylor's formula. Golfing suggestions welcome.

->n{s=t=1;n.times{s,t=t,s+t-2*(s%t)};"#{s}/#{t}"}

Python 2, 85 81 bytes

x,s=input(),[1,1]
for i in range(x):s+=s[i]+s[i+1],s[i+1]
print`s[x-1]`+"/"+`s[x]`

This submission is 1-indexed.

Using a recursive function, 85 bytes:

s=lambda x:int(x<1)or x%2 and s(x/2)or s(-~x/2)+s(~-x/2)
lambda x:`s(x)`+"/"+`s(x+1)`

If an output like True/2 is acceptable, here's one at 81 bytes:

s=lambda x:x<1 or x%2 and s(x/2)or s(-~x/2)+s(~-x/2)
lambda x:`s(x)`+"/"+`s(x+1)`

Python 2, 59 bytes

s,t=1,1;exec"s,t=t,s+t-2*(s%t);"*input();print'%d/%d'%(s,t)

Uses Peter's formula and is similarly 0-indexed.

Try it online

C#, 91 90 bytes

n=>{Func<int,int>s=_=>_;s=m=>1==m?m:s(m/2)+(0==m%2?0:s(m/2+1));return$"{s(n)}/{s(n+1)}";};

Casts to Func<int, string>. This is the recursive implementation.

Ungolfed:

n => 
{
    Func<int,int> s = _ => _; // s needs to be set to use recursively. _=>_ is the same byte count as null and looks cooler.
    s = m =>
        1 == m ? m               // s(1) = 1
        : s(m/2) + (0 == m%2 ? 0 // s(m) = s(m/2) for even
        : s(m/2+1));             // s(m) = s(m/2) + s(m/2+1) for odd
    return $"{s(n)}/{s(n+1)}";
};

Edit: -1 byte. Turns out C# doesn't need a space between return and $ for interpolated strings.

Octave, 90 bytes

function f(i)S=[1 1];for(j=1:i/2)S=[S S(j)+S(j+1) (j+1)];end;printf("%d/%d",S(i),S(i+1));end

><>, 34+2 = 36 bytes

After seeing Peter Taylor's excellent answer, I re-wrote my test answer (which was an embarrassing 82 bytes, using very clumsy recursion).

&01\$n"/"on;
&?!\:@}:{:@+@%2*-&1-:

It expects the input to be present on the stack, so +2 bytes for the -v flag. Try it online!

Python 2, 66 bytes

f=lambda n:1/n or f(n/2)+n%2*f(-~n/2)
lambda n:`f(n)`+'/'+`f(n+1)`

Uses the recursive formula.

m4, 131 bytes

define(s,`ifelse($1,1,1,eval($1%2),0,`s(eval($1/2))',`eval(s(eval(($1-1)/2))+s(eval(($1+1)/2)))')')define(r,`s($1)/s(eval($1+1))')

Defines a macro r such that r(n) evaluates according to the spec. Not really golfed at all, I just coded the formula.

JavaScript (ES6), 43 bytes

f=(i,n=0,d=1)=>i?f(i-1,d,n+d-n%d*2):n+'/'+d

1-indexed; change to n=1 for 0-indexed. The linked OEIS page has a useful recurrence relation for each term in terms of the previous two terms; I merely reinterpreted it as a recurrence for each fraction in terms of the previous fraction. Unfortunately we don't have inline TeX so you'll just have to paste it onto another site to find out how this formats:

$$ \frac{a}{b} \Rightarrow \frac{b}{a + b - 2(a \bmod b)} $$

R, 93 bytes

f=function(n)ifelse(n<3,1,ifelse(n%%2,f(n/2-1/2)+f(n/2+1/2),f(n/2)))
g=function(n)f(n)/f(n+1)

Literally the simplest implementation. Working on golfing it a bit.

Jelly, 16 bytes

L‘Hị⁸Sṭ
1Ç¡ṫ-j”/

Try it online! or verify all test cases.

How it works

1Ç¡ṫ-j”/  Main link. Argument: n

1         Set the return value to 1.
 Ç¡       Apply the helper link n times.
   ṫ-     Tail -1; extract the last two items.
     j”/  Join, separating by a slash.


L‘Hị⁸Sṭ   Helper link. Argument: A (array)

L         Get the length of A.
 ‘        Add 1 to compute the next index.
  H       Halve.
   ị⁸     Retrieve the item(s) of A at those indices.
          If the index in non-integer, ị floors and ceils the index, then retrieves
          the items at both indices.
    S     Compute the sum of the retrieved item(s).
     ṭ    Tack; append the result to A.

CJam (20 bytes)

1_{_@_2$%2*-+}ri*'/\

Online demo. Note that this is 0-indexed. To make it 1-indexed, replace the initial 1_ by T1.

Dissection

This uses the characterisation due to Moshe Newman that

the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 - x)

If x = s/t then we get

  1 / (2 * floor(s/t) + 1 - s/t)
= t / (2 * t * floor(s/t) + t - s)
= t / (2 * (s - s%t) + t - s)
= t / (s + t - 2 * (s % t))

Now, if we assume that s and t are coprime then

  gcd(t, s + t - 2 * (s % t))
= gcd(t, s - 2 * (s % t))
= gcd(t, -(s % t))
= 1

So a(n+2) = a(n) + a(n+1) - 2 * (a(n) % a(n+1)) works perfectly.

1_           e# s=1, t=1
{            e# Loop...
  _@_2$%2*-+ e#   s, t = t, s + t - 2 * (s % t)
}
ri*          e# ...n times
'/\          e# Separate s and t with a /

MATL, 32 30 bytes

1i:"tt@TF-)sw@)v]tGq)V47bG)Vhh

This uses a direct approach: generates enough members of the sequence, picks the two desired ones and formats the output.

Try it online!

MATL, 20 bytes

FT+"@:qtPXnosV47]xhh

This uses the characterization in terms of binomial coefficients given in the OEIS page.

The algorithm works in theory for all numbers, but in practice it is limited by MATL's numerical precision, and so it doesn't work for large entries. The result is accurate for inputs up to 20 at least.

Try it online!

Explanation

FT+      % Implicitly take input n. Add [0 1] element-wise. Gives [n n+1]
"        % For each k in [n n+1]
  @:q    %   Push range [0 1 ... k-1]
  tP     %   Duplicate and flip: push [k-1 ... 1 0]
  Xn     %   Binomial coefficient, element-wise. Gives an array
  os     %   Number of odd entries in that array
  V      %   Convert from number to string
  47     %   Push 47, which is ASCII for '\'
]        % End for each
x        % Remove second 47
hh       % Concatenate horizontally twice. Automatically transforms 47 into '\'
         % Implicitly display

05AB1E, 34 33 25 23 bytes

XX‚¹GÂ2£DO¸s¦ìì¨}R2£'/ý

Explanation

XX‚                        # push [1,1]
   ¹G           }          # input-1 times do
     Â                     # bifurcate
      2£                   # take first 2 elements of reversed list
        DO¸                # duplicate and sum 1st copy, s(n)+s(n+1)
           s¦              # cut away the first element of 2nd copy, s(n)
             ìì            # prepend both to list
               ¨           # remove last item in list
                 R2£       # reverse and take the first 2 elements
                    '/ý    # format output
                           # implicitly print

Try it online

Saved 2 bytes thanks to Adnan.