Parity of Nth Term in Fibonacci Integer Sequences

Question

If you look at the Fibonacci Numbers, you will notice a pattern in their parity: 0, 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144. Every third number is even, and all the others are odd. This makes sense because an even plus an odd is odd, but an odd plus an odd is even and the sum for a term will always include at least one even number unless the last two terms were odd, which happens every three terms.

In this challenge you will have to look at a generalization of this problem.

Specs

• You will be given as input two positive integers describing the Fibonacci integer sequence you will have to look at.
• For example, 0, 1 describes the original Fibonacci numbers, and 2, 1 would be the Lucas numbers. The sequence is calculated with the recurrence relation: x(n+2)=x(n+1)+x(n).
• You will also take a positive integer n.
• Your code has to output the parity of the nth term of the described Fibonacci integer sequence (0 indexed).
• The output format can be any two sane, consistent outputs (e.g. 0 for even and 1 for odd).

Test Cases

(Fibonacci numbers) [0, 1], 4  ->  odd
(Lucas numbers) [2, 1], 9  ->  even
[14, 17], 24  ->  even
[3, 9], 15  ->  odd
[2, 4], 10  ->  even

Answer 1

Python, 29 bytes

lambda a,b,n:[a,b,a+b][n%3]%2

The generalized Fibonacci sequence modulo 2 has cycle length of 3. So, we reduce n mod 3 and take that element of the first three.

---

31 bytes:

lambda a,b,n:0<a%2*2+b%2!=n%3+1

True for odd, False for even

Answer 2

Jelly, 4 bytes

^¡^Ḃ

Prints 0 for even, 1 for odd. Try it online!

How it works

The quick ¡ takes a link it applies repeated to the previous output, and optionally a second link that specifies the number of iterations. If that second link is absent (which is the case here), the number of iterations is taken from the last command-line argument.

When the first link is dyadic (which is the case with ^), ¡ applies that link to the previous return value (which is initialized as the left argument) and the right argument, then updates the return value with the result and the right argument with the previous return value. After doing so as many times as specified in the number of iterations, it returns the last return value.

The quicklink ^¡ is called with the left argument (first command-line argument) and the result of ^ (bitwise XOR of the left and right argument, i.e., the first and second command-line argument). XORing is necessary since the right argument is not among the return values, meaning that the right argument has to be the x(-1) in order to return x(n) with x(0) ^ n ¡ x(1).

To actually calculate x(n), we'd use the code +¡_@ (+¡ instead of ^¡ and _@ – left argument subtracted from right argument – instead of ^), but since we're only interested in the parities, any combination of +, _ and ^ will do.

Finally, Ḃ (bit) computes and returns the parity of x(n).

Answer 3

Python, 38 bytes

lambda a,b,n:(b*(n%3>0)+a*(~-n%3>0))%2

Answer 4

MATL, 10 7 bytes

tshwQ)o

Try it online!

Explanation:

t            #Duplicate the array of numbers
 s           #Sum them
  h          #and add that to the end of the array
   w         #Swap inputs so that the *N* is on top
    Q        #Increment *N* (Since MATL uses 1-based indexing)
     )       #Get that element of the array. MATL uses modular indexing.
      o      #Return it's parity

Answer 5

Perl 6,  24  23 bytes

{(|@^a,sum @a)[$^n%3]%2}
{(|@^a,&[+]...*)[$^n]%2}
{(|@^a,*+*...*)[$^n]%2}

Test:

use v6.c;
use Test;

constant even = 0;
constant odd = 1;
my @test = (
  ([0, 1], 4)    => odd,
  ([2, 1], 9)    => even,
  ([14, 17], 24) => even,
  ([3, 9], 15)   => odd,
  ([2, 4], 10)   => even,
);

my &seq-parity = {(|@^a,*+*...*)[$^n]%2}

for @test -> ( :key($input), :value($expected) ) {
  is seq-parity(|$input), $expected, ($input => <even odd>[$expected]).gist
}

ok 1 - ([0 1] 4) => odd
ok 2 - ([2 1] 9) => even
ok 3 - ([14 17] 24) => even
ok 4 - ([3 9] 15) => odd
ok 5 - ([2 4] 10) => even

Explanation:

{
  (
    # generate the Fibonacci integer sequence
    # ( @^a is the first two values )
    |@^a, *+* ... *
  )[ $^n ] # get the one at the appropriate index
  % 2 # modulo 2
}

Answer 6

J, 15 bytes

2|0{(]{:,+/)^:[

Simple method. Computes the n^th term, and takes it modulo 2 to find its parity. Even parity is represented by 0 and odd parity by 1.

Usage

   f =: 2|0{(]{:,+/)^:[
   4 f 0 1
1
   9 f 2 1
0
   24 f 14 17
0
   15 f 3 9
1

Explanation

2|0{(]{:,+/)^:[  Input: n on LHS, and initial values [a, b] on RHS
    ( .... )^:[  Repeat the function n times
         +/      Find the sum of a+b
      {:         Get the value b from the tail of [a, b]
        ,        Join the values to get [b, a+b]
     ]           Return those values as the new [a, b]
  0{             Select the first value at index 0 from the list (after n applications)
2|               Take that value modulo 2 and return

Answer 7

Pyke, 9 bytes

3%QDs+@2%

Try it here!

    s+    -   input+sum(input)
      @   -  ^[V]
3%        -   line_2 % 3
       2% - ^ % 2

Answer 8

Julia, 23 bytes

x\n=[x;sum(x)][n%3+1]%2

Try it online!

Answer 9

Actually, 12 bytes

╗│+k3╜%@E2@%

This uses the same strategy as xnor's Python answer. Input is a b n, with the spaces replaced by newlines. Output is 0 for even parity and 1 for odd parity.

Try it online!

Explanation:

╗│+k3╜%@E2@%
              implicit input ([a, b, n])
╗             save n in reg0 ([a, b])
 │            duplicate stack ([a, b, a, b])
  +k          add, push stack as list ([[a, b, a+b]])
    3╜%       n mod 3 ([[a, b, a+b], n%3])
       @E     element in list ([[a, b, a+b][n%3])
         2@%  mod 2 ([[a, b, a+b][n%3]%2)

Answer 10

Mathematica, 38 36 bytes

OddQ@({#,#2,+##}&@@#)[[#2~Mod~3+1]]&

Anonymous function, based off of @xnor's Python approach. Takes a list and an integer, and outputs either True for odd or False for even. Not much beyond that.

Answer 11

CJam, 9 bytes

q~_:++=2%

Even is zero, odd is one.

Try it online!

Explanation

q~        e# Read input and evaluate
  _       e# Duplicate initial tuple
   :+     e# Sum initial values
     +    e# Append sum to initial tuple
      =   e# Take the n-th element (modular indexing)
       2% e# Modulo 2