# Sum the First n Even Fibonacci Numbers

There seems not to be a contest for this one yet.

The task is simple. Add the first n numbers of the Fibonacci sequence that are even and output the result.

This is given by OEIS A099919, except that sequence is shifted by one, starting with fib(1) = 0 instead of fib(1) = 1.

This is code golf. Lowest byte count wins.

# Examples

n sum
1 0
2 2
3 10
4 44
5 188
6 798
7 3382
8 14328
9 60696


Related

• https://oeis.org/A099919
– xnor
Jan 3, 2017 at 23:42
• @EasterlyIrk The test cases imply the latter, but it should be explicitly stated.
– user45941
Jan 3, 2017 at 23:49
• @Mego yeah, I figured as much. Jan 3, 2017 at 23:49
• Please don't accept answers so fast. It's only been an hour, golfier answer could come in. EDIT: I see now there's already a shorter answer that's not accepted yet. Jan 4, 2017 at 0:42
• It's customary to wait at least a week before accepting an answer, because many people interpret it as a sign that the challenge is no longer active. Jan 4, 2017 at 7:54

## Python, 33 bytes

c=2+5**.5
lambda n:(7-c)*c**n//20


Try it online

Magic formula!

• Oh god. It took me much longer than it should have to realize why you were "commenting out" that 20 on the second line :P
– Theo
Jan 4, 2017 at 13:14
• @xnor, Any reference to this magic formula? Jan 4, 2017 at 13:43
• @TheChetan: possibly a(n) = (-10 + (5-3*sqrt(5))*(2-sqrt(5))^n + (2+sqrt(5))^n*(5+3*sqrt(5)))/20 (Colin Barker, Nov 26 2016) from the OEIS page Jan 4, 2017 at 13:50

# Oasis, 87 5 bytes

1 byte saved thanks to @ETHProductions and 2 more saved thanks to @Adnan!

zc»+U


Try it online!

### Explanation:

This uses the same recurrence formula as my MATL answer.

• Oasis's info.txt says U is replaced in the code with 00, might that save you a byte? Jan 4, 2017 at 0:06
• @ETHproductions Thanks! I forgot that Jan 4, 2017 at 0:14
• Nice! You can replace 4* with z and 2+ with » :) Jan 4, 2017 at 0:29
• @Adnan Thank you! I really should read the doc :-) Jan 4, 2017 at 0:40

# Python 2, 35 bytes

f=lambda n:n/2and 4*f(n-1)+f(n-2)+2


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# Actually, 6 bytes

r3*♂FΣ


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Explanation:

Every third Fibonacci number (starting from F_0 = 0) is even. Thus, the first n even Fibonacci numbers are F_{i*3} for i in [0, n).

r3*♂FΣ
r       [0, n)
3*     multiply each element by 3
♂F   retrieve the corresponding element in the Fibonacci sequence
Σ  sum


## JavaScript (ES6), 27 bytes

f=x=>x>1&&4*f(x-1)+f(x-2)+2


Recursion to the rescue! This uses one of the formulas on the OEIS page:

f(n < 1) = 0, f(n) = 4*a(n+1)+a(n)+2

(but shifted by one because the challenge shifts it by one)

## Pyke, 6 bytes

3m*.bs


Try it here!

3m*    -   map(i*3, range(input))
.b  -  map(nth_fib, ^)
s - sum(^)


# Perl 6,  38 35  32 bytes

{[+] grep(*%%2,(1,&[+]...*))[^($_-1)]}  Try it {[+] grep(*%%2,(0,1,*+*...*))[^$_]}


Try it

{[+] (0,1,*+*...*)[3,6...^$_*3]}  Try it ## Expanded: { # bare block lambda with implicit parameter ｢$_｣

[+]                       # reduce with ｢&infix:<+>｣

( 0, 1, * + * ... * )\  # fibonacci sequence with leading 0

[ 3, 6 ...^ $_ * 3 ] # every 3rd value up to # and excluding the value indexed by # the input times 3 }  # Octave, 3635 33 bytes @(n)filter(2,'FAD'-69,(1:n)>1)(n)  Try it online! ### Explanation This anonymous function implements the difference equation a(n) = 4*a(n-1)+a(n-2)+2 as a recursive filter: Y = filter(B,A,X) filters the data in vector X with the filter described by vectors A and B to create the filtered data Y. The filter is a "Direct Form II Transposed" implementation of the standard difference equation: a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb) - a(2)*y(n-1) - ... - a(na+1)*y(n-na) In our case A = [1 -4 -1], B = 2, and the input x should be a vector of ones, with the result appearing as the last entry of the output y. However, we set to 0 the first value of the input so that an initial 0 appears in the output, as required. 'FAD'-69 is just a shorter way to produce the coefficient vector A = [1 -4 -1]; and (1:n)>1 produces the input vector x = [0 1 1 ... 1]. # dc, 25 22 bytes 9k5v1+2/3?*1-^5v/0k2/p  Try it online! Or save the program in a file and run it by typing dc -f *filename*  The program accepts a non-negative integer n on stdin, and it outputs the sum of the first n even Fibonacci numbers on stdout. (The Fibonacci sequence is taken to start with 0, as per the OP's examples.) This program uses the formula (F(3n-1)-1)/2 for the sum of the first n even Fibonacci numbers, where F is the usual Fibonacci function, given by F(0) = 0, F(1) = 1, F(n) = F(n-2) + F(n-1) for n >= 2. dc is a stack-based calculator. Here's a detailed explanation: 9k # Sets the precision to 9 decimal places (which is more than sufficient). 5v # Push the square root of 5 1+ # Add 1 to the number at the top of the stack. 2/ # Divide the number at the top of the stack by 2.  At this point, the number (1+sqrt(5))/2 is at the top of the stack. 3 # Push 3 on top of the stack. ? # Read a number from stdin, and push it. \* # Pop two numbers from the stack, multiply them, and push the product 1- # Subtract 1 from the number at the top of the stack.  At this point, 3n-1 is at the top of the stack (where n is the input), and (1+sqrt(5))/2 is second from the top. ^ # Pop two numbers from the stack (x, then y), compute the power y^x, and push that back on the stack. 5v/ # Divide the top of the stack by sqrt(5).  At this point, the number at the top of the stack is (((1+sqrt(5))/2)^(3n-1))/sqrt(5). The closest integer to this number is F(3n-1). Note that F(3n-1) is always an odd number. 0k # Change precision to 0 decimal places. 2/ # Divide the top of the stack by 2, truncating to an integer. p # Print the top of the stack on stdout.  ## Mathematica, 27 21 bytes Thanks to xnor for pointing out an alternate formula, alephalpha for correcting for starting index Fibonacci[3#-1]/2-.5&  • Might the (Fibonacci(3*n+2)-1)/2 formula be shorter? – xnor Jan 4, 2017 at 0:38 # Forth (gforth), 38 bytes : f 1 1 rot 1 ?do tuck 4 * + loop 2/ ;  Try it online! Wins over existing Forth answer by porting this formula: $$G(0)=G(1)=1, G(n+2)=4G(n+1)+G(n) \\ a(n) = \left\lfloor \frac{G(n)}2 \right\rfloor$$ : f ( n -- a_n ) \ Take single number n and return a_n 1 1 rot 1 \ g0 g1 n 1 ?do \ loop n-1 times tuck 4 * + \ gn-1 gn -> gn gn+1=4gn+gn-1 loop \ end loop; top is gn 2/ \ an = gn/2 (floor division) ;  # MATL, 15 14 bytes OOi:"t4*b+2+]x  Try it online! ### Explanation This uses one of the recurrence formulas from OEIS: a(n) = 4*a(n-1)+a(n-2)+2 For input N the code iterates N times, which is 2 more times than necessary. This is compensated for by setting 0, 0 (instead of 0, 2) as initial values, and by deleting the last obtained value and displaying the previous one. OO % Push two zeros as initial values of a(n-2), a(n-1) i % Input N :" % Do this N times t % Duplicate a(n-1) 4* % Multiply by 4 b+ % Bubble up a(n-2) and add to 4*a(n-1) 2+ % Add 2. Now we have 4*a(n-1)+a(n-2)+2 as a(n), on top of a(n-1) ] % End x % Delete last value, a(n). Implicitly display the remaining value, a(n-1)  ## Batch, 80 bytes @set/at=x=0,y=1 @for /l %%i in (2,1,%1)do @set/az=x+y,y=z+x,t+=x=y+z @echo %t%  Uses the fact that every third Fibonacci number is even, and just calculates them three at a time (calculating more than one at a time is actually easier as you don't have to swap values around). I tried the (Fibonacci(3*n+2)-1)/2 formulation but it's actually a few bytes longer (t+= turns out to be quite efficient in terms of code size). # C, 8238 36 bytes 2 bytes saved thanks to @BrainSteel The formulas at the OEIS page made it much shorter: a(n){return--n<1?0:4*a(n)+a(n-1)+2;}  Try it online! 82 bytes: x,s,p,n,c;f(N){s=0;p=n=1;c=2;while(n<N){if(~c&1)s+=c,n++;x=p+c;p=c;c=x;}return s;}  The first version is 75 bytes but the function is not reusable, unless you always call f with greater N than the previous call :-) x,s,p=1,n=1,c=2;f(N){while(n<N){if(~c&1)s+=c,n++;x=p+c;p=c;c=x;}return s;}  My first answer here. Didn't check any other answers nor the OEIS. I guess there are a few tricks that I can apply to make it shorter :-) • You can make this a tad shorter by shuffling things around a bit: a(n){return--n<1?0:4*a(n)+a(n-1)+2;} (36 bytes) Jan 5, 2017 at 21:30 # APL (Dyalog Unicode), 22 21 bytes -1 byte thanks to Adám (2+5*.5)∘(⌊20÷⍨*×7-⊣)  Try it online! (2+5*.5)∘(⌊20÷⍨*×7-⊣) (2+5*.5) ⍝ constant c ∘( ) ⍝ joined to 7-⊣ ⍝ 7-c × ⍝ multiplied by * ⍝ input^c 20÷⍨ ⍝ all over 20 ⌊ ⍝ floor  # Or with APL (Dyalog Extended), 19 bytes (Thanks to Adám) (2+√5)∘(⌊20÷⍨*×7-⊣)  Try it online! • Save a byte by answering with the tacit function (2+5*.5)∘(⌊20÷⍨*×7-⊣) Try it online! and optionally another two from Dyalog Extended's √ Try it online! – Adám Feb 11, 2020 at 9:46 • Why is your bounty request for 50 instead of 100 (doubled by being well-explained)? – Adám Feb 11, 2020 at 9:50 • I didn't know what would count for well explained and is it is just a re-implementation of someone else's formula, I didn't think think it would count. @Adám Feb 11, 2020 at 16:02 # APL (Dyalog Unicode), 16 bytesSBCS ⌊⊃⌽(4∘⊥,⊃)⍣⎕÷2 2  Try it online! A full program. ### How it works: the math Uses a(n) = (Fibonacci(3*n + 2) - 1)/2 from the OEIS page, but we have offset 1 in a(n), so the actual formula is: $$a(n) = \frac{F(3 n - 1) - 1}{2} = \Bigl\lfloor \frac{F(3 n - 1)}{2} \Bigr\rfloor$$ And using $$\ F(2) = F(-1) = 1 \$$ and $$\ F(n+3) = 4F(n) + F(n-3) \$$, we define a derived sequence $$\ G(n) \$$ such that $$G(0)=G(1)=\frac12, G(n+2)=4G(n+1)+G(n)$$ Then the desired value is $$\ a(n) = \lfloor G(n) \rfloor \$$. ### How it works: the code ⌊⊃⌽(4∘⊥,⊃)⍣⎕÷2 2 ⍝ Input: n ÷2 2 ⍝ 2-element vector [.5 .5] i.e. [G(1) G(0)] (4∘⊥,⊃)⍣⎕ ⍝ Derive [G(x+2) G(x+1)] from [G(x+1) G(x)] n times: 4∘⊥ ⍝ G(x+2) = 4×G(x+1) + G(x) ⊃ ⍝ G(x+1) as first element of the vector , ⍝ Concatenate ⊃⌽ ⍝ Get the last element of the result vector, i.e. G(n) ⌊ ⍝ Floor  ## Haskell (32 31 bytes) Saved one byte thanks to @ChristianSievers. Using the formula given in OEIS: a(n) = 4*a(n-1)+a(n-2)+2, n>1 by Gary Detlefs a n|n>1=4*a(n-1)+a(n-2)+2|n<2=0 • A golfier way to say n<=1 for integers is n<2. Also, the second condition doesn't need to be the negation of the first (the idiomatic otherwise is simply True), so usally in golfing something like 1<2 is used. Jan 4, 2017 at 12:44 • @ChristianSievers indeed the n<2 is an obvious improvement, thank you. The second one works as well, though it does not save me anything in this case. I'm still learning Haskell and did not realise I could have a guard like that. Thank you! Jan 4, 2017 at 12:49 ## Mathematica, 32 27 bytes Fibonacci[3Input[]-1]/2-1/2  Credit to xnor. Saved 5 bytes thanks to JungHwan Min. • Surely Mathematica has Fibonacci and it's shorter to do either (Fibonacci(3*n+2) - 1)/2 or write the sumi? – xnor Jan 4, 2017 at 0:19 • @JungHwanMin This isn't plagiarism; it mentions the OEIS page. Also, this isn't a candidate for community wiki. See How should Community Wikis be used?. Jan 4, 2017 at 18:13 • @devRichter Sorry for undeleting your post, but it was necessary to have a conversation. If you want to keep it deleted, let me know and I'll move this conversation to a chat room. Jan 4, 2017 at 18:15 • @Dennis still, I believe credit should be given to Vincenzo Librandi explicitly -- (accidentally deleted my last comment... could that be undeleted?) For the community post suggestion, I stand corrected. Jan 4, 2017 at 18:16 • What I meant was to mention his name in the post... (or perhaps include the Mathematica comment (* Vincenzo Librandi, Mar 15 2014 *) in the post, as it is on OEIS.) Jan 4, 2017 at 18:24 # R, 42 bytes Non-recursive solution, as contrast to the earlier solution by @rtrunbull here. for(i in 1:scan())F=F+gmp::fibnum(3*i-3);F  Uses the property that each third value of the Fibonacci sequence is even. Also abuses the fact that F is by default defined as FALSE=0, allowing it as a basis to add the values to. # R, 42 41 bytes sum(DescTools::Fibonacci(3*(scan():2-1)))  scan() : take n from stdin. scan():2-1 : generate integers from n to 2, decrement by 1, yielding n-1 through 1. 3*(scan():2-1) : multiply by 3, as every third fibonacci number is even. DescTools::Fibonacci(3*(scan():2-1)) : Return these fibonacci numbers (i.e. 3 through (n-1)*3). sum(DescTools::Fibonacci(3*(scan():2-1))) : Sum the result. Previously, I had this uninteresting solution using one of the formulae from OEIS: a=function(n)if(n<2,0,4*a(n-1)+a(n-2)+2)  • I managed to match your bytecount without recursion :) – JAD Jan 5, 2017 at 14:41 • @JarkoDubbeldam Nice! I've ditched the recursion also and made a one-byte improvement :) Jan 5, 2017 at 15:30 • Nice, what exactly does desctools::fibonacci do that numbers::fibonacci cant? Because that mist be a bit shorter. – JAD Jan 5, 2017 at 15:39 • Oh nevermind, found it. Sweet, the other implementations I found don't support asking for multiple numbers at once. – JAD Jan 5, 2017 at 15:40 • @JarkoDubbeldam Yeah, gmp::fibnum'' returns objects of type bigz, which the *apply class of functions converts to type raw because reasons... Jan 5, 2017 at 15:50 ## Japt, 10 bytes Uo@MgX*3Ãx  Try it online! Thanks ETHproductions :) # PHP, 73 70 bytes for(${0}=1;$i++<$argv[1];$$x={0}+{1}){x^=1}&1?i--:s+=$$x;echo$s;  showcasing variable variables. O(n). Run with -nr. breakdown for(${0}=1;         # init first two fibonaccis (${1}=NULL evaluates to 0 in addition) # the loop will switch between$0 and $1 as target.$i++<$argv[1]; # loop until$i reaches input
$$x={0}+{1} # 3. generate next Fibonacci ) {x^=1} # 1. toggle index (NULL => 1 => 0 => 1 ...) &1?i-- # 2. if current Fibonacci is odd, undo increment :s+=$$x;           #    else add Fibonacci to sum
echo$s; # print result  Numbers are perfectly valid variable names in PHP. But, for the literals, they require braces; i.e. ${0}, not $0. 36 bytes, O(1) <?=(7-$c=2+5**.5)*$c**$argv[1]/20|0;


# Forth (gforth), 47 bytes

: f 3 * 1- 0 1 rot 0 do tuck + loop d>s 1- 2/ ;


Try it online!

Uses the first formula for OEIS a(n) = (Fibonacci(3*n + 2) - 1)/2, since the code for Fibonacci is relatively small in forth.

: f              \ start a new word definition
3 * 1-         \ multiply by 3 and subtract 1 (simplified form of 3*(n-1) + 2 )
0 1 rot 0      \ create starting Fibonacci values and set up loop parameters
do tuck + loop \ Fibonacci code - iteratively copy top value add top 2 stack values
d>s            \ drop the top stack value
1- 2/          \ subtract 1 and divide result by 2
;                \ end word definition


# Husk, 8 bytes

Σ↑Θnİ0İf


Try it online!

painfully slow due to intersecting two infinite lists.

• Nice. This is another, rather faster, 8-byter, but obviously golf doesn't care about speed! Nov 2, 2020 at 16:33

# PARI/GP, 21 bytes

n->fibonacci(3*n-1)\2


\ is the integer quotient.

# Excel (Version 1911), 119 Bytes

Using iterative calculations (Maximum iteration 1024)

A1 'Input
B1 =IF(B1=4^5,1,B1+1)
C1 =SEQUENCE(A1*3-1)-1
D1 =IF(B1=1,N(C1#>0),IF(C1#=B1,SUM(INDEX(D1#,B1-{0,1})),D1#))
E1 =SUM(D1#*(MOD(D1#,2)=0)) 'Output


Tests

# GolfScript, 46 bytes

(:m;5 2 -1??:i 3*5\- 2i-m?*3i*5+2i+m?*+-10+20/


Ignore the floating point, TIO isn't perfect.

Yeah, yeah. I cheated. This is just a fancy arithmetic equation to calculate the result. Works in O(1) time, though, assuming a^n can be calculated in O(1) time, which might not be true.

(:m;5 2 -1??:i 3*5\- 2i-m?*3i*5+2i+m?*+-10+20/ #Sum the first n even F
(:m;                                           #Set m to our input minus one, pop the value from the list.
5 2 -1??:i                                 #Set i to sqrt(5), but don't pop it since we're gonna use it (saves 1 byte)
3*5\-                           #5-3sqrt(5)
2i-m?                     #[2-sqrt(5)]^m
*                    #Multiply those two together
3i*5+2i+m?*         #Same as the above lines, but + instead of -.
+        #Add the left part with the right part
-10+    #Add -10 (-10+20/ and 10- 20/ are the same size)
20/ #Divide by twenty.



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# 05AB1E, 7 bytes

L<3*ÅfO


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# GolfScript, 15 bytes

0.@{.4*@2++}\*;


This one actually does the "operation" using a neat little trick I found. The sum is actually just a 0, 0 leading sequence, with a(n) = 4a(n-1) + a(n-2) + 2. So I do that.

0.@{.4*@2++}\*; #Print the sum of the first n even F
0.              #Add two 0s to the stack. [n 0 0]
@             #Bring the n up front. [0 0 n]
{       }    #A block of "do something" [0 0 n {}]
{       }\   #Bring n to the front again [0 0 {} n]
{       } *  #Perform the block n times on the stack below it [0 0]
{.      }    #Duplicate our top element (a[n-1] a[n-1])
{ 4*    }    #Multiply the dupe by 4 (a[n-1] 4a[n-1])
{   @   }    #Bring up our bottom element (a[n-1] 4a[n-1] a[n-2])
{    2++}    #4a[n-1]+a[n-2]+2 = a[n]
{       }    #Stack is now a[n-1] a[n], so we dop the loop until our n is right!
; #The stack has a[n-1] a[n], but our "n" in question is 1 too high
#So we pop the top element, leaving a[n-1], W5.


Try it online!

# Jelly, 6 bytes

Ḷ×3ÆḞS


Try it online!

Uses the formula

$$a(n) = \sum_{i=0}^{n-1}f(3i)$$

where $$\f(n)\$$ is the $$\n\$$-th Fibonacci number

## How it works

Ḷ×3ÆḞS - Main link. Take n on the left
Ḷ      - [0, 1, 2, ..., n-1]
×3    - [0, 3, 6, ..., 3n-3]
ÆḞ  - n'th Fibonacci number for each
S - Sum