JavaScript (ES6), 45 bytes

f=(n,x=0,y=1)=>n?f(n-1,y,(x%9||x)+(y%9||y)):x

<input type=number min=0 oninput=o.textContent=f(this.value)><pre id=o>

x and y can't both be 9, since that would require the previous number to be 0, so their digital sum can't exceed 17. This means that the digital root for numbers greater than 9 is the same as the remainder modulo 9.

Python 2, 53 bytes

f=lambda n:n>1and sum(map(int,`f(n-1)`+`f(n-2)`))or n

Recursive function. The base cases of n=0 and n=1 yield n, larger numbers calculate the value by calling f(n-1) and f(n-2) converting each to a string, concatenating the two strings, converting each character to an integer using a map with the int function, and then sums the resulting list.

Using the modulo-24 information I can currently get a 56 byte non-recursive unnamed function:

lambda n:int(('011'+'2358dc7a89hhgfda56b8a9aa'*n)[n],18)

JavaScript (ES6), 34 bytes

f=n=>n<2?n:~-f(--n)%9+~-f(--n)%9+2

May freeze your browser for inputs above 27 or so, but it does work for all input values. This can be verified with a simple cache:

c=[];f=n=>n<2?n:c[n]=c[n]||~-f(--n)%9+~-f(--n)%9+2

<input type=number value=0 min=0 step=1 oninput="O.value=f(this.value)"> <input id=O value=0 disabled>

As pointed out in Neil's brilliant answer, the output can never exceed 17, so the digital sum of any output above 9 is equal to n%9. This also works with outputs below 9; we can make it work for 9 as well by subtracting 1 with ~- before the modulus, then adding back in 1 after.

The best I could do with hardcoding is 50 bytes:

n=>"0x"+"7880136ba5867ffedb834968"[n%24]-(n<3)*9+2

Jelly, 8 bytes

;DFS
ç¡1

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How it works

ç¡1   Main link. No arguments. Implicit left argument: 0

  1   Set the right argument to 1.
ç¡    Repeatedly execute the helper link n times – where n is an integer read from
      STDIN – updating the left argument with the return value and the right
      argument with the previous value of the left argument. Yield the last result.


;DFS  Helper link. Arguments: a, b

;     Concatenate; yield [a, b].
 D    Decimal; convert both a and b to their base-10 digit arrays.
  F   Flatten the result.
   S  Compute the sum of the digits.

Alternate solution, 19 bytes, constant time

;DFS
9⁵ç23С⁸ịµṠ>?2

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How it works

9⁵ç23С⁸ịµṠ>?2  Main link. Argument: n

9               Set the return value to 9
 ⁵              Yield 10.
  ç23С         Execute the helper link 23 times, with initial left argument 10
                and initial right argument 9, updating the arguments as before.
                Yield all intermediate results, returning
                [10,10,2,3,5,8,13,12,7,10,8,9,17,17,16,15,13,10,5,6,11,8,10,9].
   ⁸ị           Extract the element at index n. Indexing is 1-based and modular.
     µ          Combine all links to the left into a chain.
       >?2      If n > 2, execute the chain.
      Ṡ         Else, yield the sign if n.

JavaScript (ES6), 52 46 45 bytes

_=$=>$<2?$:eval([..._(--$)+[_(--$)]].join`+`)

Usage

_=$=>$<2?$:eval([..._(--$)+[_(--$)]].join`+`)
_(7)

Output

13

Explanation

This function checks if the input is smaller than 2, and if so, it returns the input. Otherwise, it creates an array of two values that are appended to each other as strings. Those two values are the result of calling the function with input - 1 and input - 2.

The ... operator splits this string into an array of characters, which then is converted to a string again, but now with +es between values. This string is then interpreted as code, so the sum is calculated, which is then returned.

This is a double-recursive algorithm, which makes it quite inefficient. It needs 2^n-2 function calls for input n. As such, here's a longer but faster solution. Thanks to ETHproductions for coming up with it.

f=($,p=1,c=0)=>$?f($-1,c,eval([...p+[c]].join`+`)):c

05AB1E, 8 bytes

XÎF‚¤sSO

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Explanation

XÎ        # push 1,0,input
  F       # input_no times do:
   ‚      # pair the top 2 elements of the stack
    ¤     # push a copy of the 2nd element to the stack
     s    # swap the pair to the top of the stack
      S   # convert to list of digits
       O  # sum

Mathematica, 49 bytes

If[#<2,#,Tr[Join@@IntegerDigits[#0/@{#-1,#-2}]]]&

Straightforward recursive definition. Gets pretty slow after a while.

Mathematica, 79 bytes

If[#<3,Sign@#,(9@@43626804920391712116157158790~IntegerDigits~18)[[#~Mod~24]]]&

Hardcoding the periodic pattern. Lightning fast and satisfyingly abusive to Mathematica :)

MATL, 15 bytes

lOi:"yyhFYAss]&

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lO       % Push 1, then 0. So the next generated terms will be 1, 1, 2,... 
i        % Input n
:"       % Repeat that many times
  yy     %   Duplicate top two elements in the stack
  h      %   Concatenate into length-2 horizontal vector
  FYA    %   Convert to decimal digits. Gives a 2-row matrix
  ss     %   Sum of all matrix entries
]        % End
&        % Specify that next function (display) will take only 1 input
         % Implicit display

CJam, 22 20 bytes

Saved 2 bytes thanks to Martin Ender

ri2,{(_(jAb\jAb+:+}j

Straightforward recursive algorithm, nothing fancy. 0-indexed.

Try it online! or test for 0-50 (actually runs pretty fast).

Explanation

ri                    Read an integer from input
  2,                  Push the array [0 1]
    {             }j  Recursive block, let's call it j(n), using the input as n and [0 1] as base cases
     (                 Decrement (n-1)
      _(               Duplicate and decrement again (n-2)
        jAb            Get the list digits of j(n-2)
           \           Swap the top two elements
            jAb        Get the list of digits of j(n-1)
               +       Concatenate the lists of digits
                :+     Sum the digits

CJam, 42 bytes

Solution using the repetition. Similar algorithm to Jonathan Allan's solution.

ri_2,1+"[2358DC7A89HHGFDA56B8A9AA]"S*~@*+=

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Python 2, 56 bytes

Simple iterative solution.

a,b=0,1
exec'a,b=b,(a%9or a)+(b%9or b);'*input()
print a

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Using (a%9or a)+(b%9or b) actually turned out to be shorter than sum(map(int(`a`+`b`)))!

PowerShell, 79 bytes

$b,$c=0,1;for($a=$args[0];$a;$a--){$z=[char[]]"$b$c"-join'+'|iex;$b=$c;$c=$z}$b

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Lengthy boring iterative solution that does direct digit-sum calculations each for loop. At the end of the loop, the number we want is in $b, so that's left on the pipeline and output is implicit. Note that if the input is 0, then the loop won't enter since the conditional is false, so $b remains 0.

Python 2, 54 46 bytes

f=lambda n:n>2and~-f(n-1)%9+~-f(n-2)%9+2or n>0

Heavily inspired by @ETHproductions' ES6 answer.

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Perl 6, 41 bytes

{(0,1,{[+] |$^a.comb,|$^b.comb}...*)[$_]}

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{ # bare block lambda with implicit parameter ｢$_｣
  (

    0, 1,           # first two values

    {
      [+]          # sum
        |$^a.comb, # the digits of the first parameter (n-2)
        |$^b.comb  # the digits of the second parameter (n-1)
    }

    ...            # keep using that code block to generate new values until

    *              # never stop

  )[ $_ ]          # index into the sequence
}

Batch, 85 bytes

@set/ax=0,y=1
@for /l %%i in (1,1,%1)do @set/az=x-x/10*9+y-y/10*9,x=y,y=z
@echo %x%

I was wondering how I was going to port my JavaScript answer to batch but the clue was in @Dennis's Python solution.

Pyth, 20 bytes

J,01VQ=+JssjRT>2J)@J

A program that takes input of a zero-indexed integer and prints the result.

Test suite (First part for formatting)

How it works

[Explanation coming later]

Oasis, 5 bytes

Code:

ScS+T

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Expanded version:

ScS+10

Explanation:

ScS+   = a(n)
     0 = a(0)
    1  = a(1)
S      # Sum of digits on a(n-1)
 c     # Compute a(n-2)
  S    # Sum of digits
   +   # Add together

R, 90 bytes

A horribly long solution, but it's better than the 108 I originally had. I suspect that there is a lot better way to do this, but I can't see it at the moment.

function(n,x=0:1){repeat`if`(n,{x=c(x,sum(scan(t=gsub('',' ',x))))[-1];n=n-1},break);x[1]}

This is an unnamed function that uses gsub and scan(t= to split the numbers in the vector into digits. The sum of these is added to the vector while the first item is dropped. repeat is used to step through the sequence n times and the result is the first item of the vector.

C, 96 bytes

or 61 bytes counting escape codes as 1 byte each

0 indexed. Similar to some of the other answers I am indexing a lookup table of values but I have compressed it into 4 byte chunks. I didn’t bother investigating the mod 24 version because I didn’t think it was as interesting since the others have done so already but let’s face it, C isn’t going to win anyway.

#define a(n) n<3?!!n:2+(15&"\x1\x36\xba\x58\x67\xff\xed\xb8\x34\x96\x87\x88"[(n-3)/2%12]>>n%2*4)

explanation:

#define a(n)                                                                                     // using a preprocessor macro is shorter than defining a function
             n<3?!!n:                                                                            // when n is less than 3 !!n will give the series 0,1,1,1..., otherwise..
                                                                             (n-3)/2%12          // condition the input to correctly index the string...
                           "\x1\x36\xba\x58\x67\xff\xed\xb8\x34\x96\x87\x88"                     // which has the repeating part of the series encoded into 4 bits each number
                                                                                                 // these are encoded 2 less than what we want as all numbers in the series after the third are 2 <= a(n>2) <= 17 which conforms to 0 <= a(n>2) - 2 <= 15
                                                                                        >>n%2*4  // ensure the data is in the lower 4 bits by shifting it down, n%2 will give either 0 or 1, which is then multiplied by 4
                        15&                                                                      // mask those bits off
                     2+                                                                          // finally, add 2 to correct the numbers pulled from the string

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Ruby, 58 bytes

->n{n<3?n<=>0:"9aa2358dc7a89hhgfda56b8a"[n%24].to_i(18)}

The simple hardcoded solution.

Japt, 27 bytes

U<2?U:~-ß--U %9+~-ß--U %9+2

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Haskell, 76 74 bytes

import Data.Char
s=sum.map digitToInt.show
f n|n<2=n|n>1=s(f$n-1)+s(f$n-2)

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