A079345 -id:A079345 - cp's OEIS frontend

A079343 Period 6: repeat [0, 1, 1, 2, 3, 1]; also F(n) mod 4, where F(n) = A000045(n).

0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1

Offset: 0

Jon Perry, Jan 04 2003

This sequence shows that every sixth Fibonacci number (A134492) is divisible by 4. - Alonso del Arte, Jul 27 2013

			a(5) = F(5) mod 4 = 5 mod 4 = 1.
a(6) = F(6) mod 4 = 8 mod 4 = 0.
a(7) = F(7) mod 4 = 13 mod 4 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jon Maiga, Upper bound of Fibonacci entry points, (2019).
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A000045, A079344, A079345, A134492.

[Fibonacci(n) mod 4: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

A079343:=n->[0, 1, 1, 2, 3, 1][(n mod 6)+1]: seq(A079343(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

PadLeft[{}, 108, {0, 1, 1, 2, 3, 1}] (* Harvey P. Dale, Aug 10 2011 *)
Table[Mod[Fibonacci[n], 4], {n, 0, 127}] (* Alonso del Arte, Jul 27 2013 *)
LinearRecurrence[{1, -1, 1, -1, 1},{0, 1, 1, 2, 3},105] (* Ray Chandler, Aug 27 2015 *)

for (n=0,100,print1(fibonacci(n)%4","))

a(n) = 2^(1 - P(3, n) + P(6, n+2))*3^P(6, n+3) - 1, where P(k, n) = floor(1/2*cos(2*n*Pi/k) + 1/2). [Gary Detlefs, May 16 2011]
a(n) = 4/3 - cos(Pi*n/3) - sin(Pi*n/3)/sqrt(3) - cos(2*Pi*n/3)/3 + sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
G.f.: x*(1+2*x^2+x^3) / ( (1-x)*(1-x+x^2)*(1+x+x^2) ). - R. J. Mathar, Jul 14 2012
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*(2*exp(x) - sqrt(3)*sin(sqrt(3)*x/2)*sinh(x/2) - cos(sqrt(3)*x/2)*(sinh(x/2) + 2*cosh(x/2)))/3. - Ilya Gutkovskiy, Jun 20 2016

A079344 F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.

Original entry on oeis.org

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, 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, 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

Offset: 0

Jon Perry, Jan 04 2003

This sequence does not contain the complete set of residues modulo 8. See A079002. - Michel Marcus, Jan 31 2020

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Brandon Avila and Yongyi Chen, On Moduli For Which the Lucas Numbers Contain a Complete Residue System, Fibonacci Quart. 51 (2013), no. 2, 151-152. See p. 151.
S. A. Burr, On moduli for which the Fibonacci sequence contains a complete system of residues, The Fibonacci Quarterly, 9.5 (1971), 497-504.
P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.
Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1,0,0,-1,1).

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

[Fibonacci(n) mod 8: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

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 *)

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

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

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

Edited by N. J. A. Sloane, Dec 06 2008 at the suggestion of R. J. Mathar

A137290 Fibonacci(n) mod 30.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 24, 23, 17, 10, 27, 7, 4, 11, 15, 26, 11, 7, 18, 25, 13, 8, 21, 29, 20, 19, 9, 28, 7, 5, 12, 17, 29, 16, 15, 1, 16, 17, 3, 20, 23, 13, 6, 19, 25, 14, 9, 23, 2, 25, 27, 22, 19, 11, 0, 11, 11, 22, 3, 25, 28, 23, 21, 14, 5, 19, 24, 13, 7, 20, 27, 17, 14

Offset: 1

Aaron M. Churchill (churchil(AT)math.udel.edu), Mar 15 2008

nonn

Has period 120.

Michel Marcus, Table of n, a(n) for n = 1..130
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000045, A003893, A007887, A079345, A105471.

Mod[Fibonacci[Range[80]],30] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = fibonacci(n) % 30 \\ Michel Marcus, Jun 12 2013

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A079343 Period 6: repeat [0, 1, 1, 2, 3, 1]; also F(n) mod 4, where F(n) = A000045(n).

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A079344 F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.

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A137290 Fibonacci(n) mod 30.

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