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F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.
10

%I #46 Dec 14 2023 05:07:36

%S 0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,5,2,

%T 7,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,

%U 5,2,7,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,5,2,7,1,0,1,1,2,3,5,0,5,5

%N F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.

%C This sequence does not contain the complete set of residues modulo 8. See A079002. - _Michel Marcus_, Jan 31 2020

%H Vincenzo Librandi, <a href="/A079344/b079344.txt">Table of n, a(n) for n = 0..1000</a>

%H Brandon Avila and Yongyi Chen, <a href="https://www.fq.math.ca/Papers1/51-2/AvilaChen.pdf">On Moduli For Which the Lucas Numbers Contain a Complete Residue System</a>, Fibonacci Quart. 51 (2013), no. 2, 151-152. See p. 151.

%H S. A. Burr, <a href="https://www.fq.math.ca/Scanned/9-5/burr.pdf">On moduli for which the Fibonacci sequence contains a complete system of residues</a>, The Fibonacci Quarterly, 9.5 (1971), 497-504.

%H P. Ribenboim, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-1.pdf">FFF (Favorite Fibonacci Flowers)</a>, Fib. Q. 43 (No. 1, 2005), 3-14.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,-1,1,0,0,-1,1).

%F Sequence is periodic with Pisano period 12 = A001175(8).

%F G.f.: -x*(1+x^2+x^3+3*x^4+6*x^6-5*x^5+x^7) / ( (x-1)*(x^2-x+1)*(1+x+x^2)*(x^4-x^2+1) ). - _R. J. Mathar_, Aug 08 2012

%e a(8) = F(8) mod 8 = 21 mod 8 = 5.

%t Mod[Fibonacci[Range[0,110]],8] (* or *) LinearRecurrence[ {1,0,0,-1,1,0,0,-1,1},{0,1,1,2,3,5,0,5,5},110] (* _Harvey P. Dale_, Jan 16 2014 *)

%o (PARI) for (n=0,100,print1(fibonacci(n)%8","))

%o (Magma) [Fibonacci(n) mod 8: n in [0..100]]; // _Vincenzo Librandi_, Feb 04 2014

%Y Cf. A000045, A011655, A082115, A079343, A082116, A082117, A079344, A079345, A111958, A079002.

%K nonn,easy

%O 0,4

%A _Jon Perry_, Jan 04 2003

%E Edited by _N. J. A. Sloane_, Dec 06 2008 at the suggestion of _R. J. Mathar_