A082115 Fibonacci sequence (mod 3).
0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Fibonacci Number
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).
Programs
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Magma
[Fibonacci(n) mod 3: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014
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Mathematica
Table[Mod[Fibonacci[n], 3], {n, 0, 100}](* Vincenzo Librandi, Feb 04 2014 *) LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 1, 2, 0, 2, 2, 1},103] (* Ray Chandler, Aug 27 2015 *) -
PARI
a(n)=fibonacci(n%8)%3 \\ Charles R Greathouse IV, Sep 28 2015
Formula
Sequence is periodic with Pisano period 8.
a(n) = 1-floor(n/8)+floor((n-1)/8)+floor((n-3)/8)-2*floor((n-4)/8) +2*floor((n-5)/8)-floor((n-7)/8). - Hieronymus Fischer, Jul 01 2007
a(n) = 1+((n mod 8)+((n+1)mod 8)-2*((n+3)mod 8)+2*((n+4)mod 8) -((n+5)mod 8) -((n+7)mod 8))/8. - Hieronymus Fischer, Jul 01 2007
G.f.: (x+x^2+2x^3+2x^5+2x^6+x^7)/(1-x^8). - Hieronymus Fischer, Jul 01 2007
a(n) = A131295(n) mod 3 (for n>0). - Hieronymus Fischer, Jul 01 2007
Extensions
Added a(0)=0. - Jon Perry, Sep 15 2013