A082116 Fibonacci sequence (mod 5).

0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1

Offset: 0

Author

Eric W. Weisstein, Apr 03 2003

Keywords

• nonn
• easy

Comments

This sequence contains the complete set of residues modulo 5. See A079002. - Michel Marcus, Jan 31 2020

References

• S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989. See p. 88. - N. J. A. Sloane, Feb 20 2013

Links

• 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.
• Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 11.
• Minjia Shi and Patrick Solé, The largest number of weights in cyclic codes, arXiv:1807.08418 [cs.IT], 2018.
• Eric Weisstein's World of Mathematics, Fibonacci Number
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1,1,1,-1,-1,1,0,-1,0,1).

Crossrefs

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

Programs

Magma

[Fibonacci(n) mod 5: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

Mathematica

Table[Mod[Fibonacci[n], 5], {n, 0, 125}] (* Alonso del Arte, Jul 29 2013 *)

PARI

a(n)=fibonacci(n)%5 \\ Charles R Greathouse IV, Oct 07 2015

Formula

Sequence is periodic with Pisano period 20.

a(n) = 2 + ((n mod 20) - ((n - 1) mod 20) - ((n - 3) mod 20) - ((n - 4) mod 20) + 3*((n - 5) mod 20) - 3*((n - 6) mod 20) + 2*((n - 8) mod 20) - 3*((n - 9) mod 20) + 4*((n - 10) mod 20) - 4*((n - 11) mod 20) + ((n - 13) mod 20) + ((n - 14) mod 20) + 2*((n - 15) mod 20) - 2*((n - 16) mod 20) - 2*((n - 18) mod 20) + 3*((n - 19) mod 20))/20. - Hieronymus Fischer, Jun 30 2007

G.f.: (x + x^2 + 2x^3 + 3x^4 + 3x^6 + 3x^7 + x^8 + 4x^9 + 4x^11 + 4x^12 + 3x^13 + 2x^14 + 2x^16 + 2x^17 + 4x^18 + x^19)/(1 - x^20), not reduced. - Hieronymus Fischer, Jun 30 2007

a(n) = A010073(n) mod 5. - Hieronymus Fischer, Jun 30 2007

G.f.: -x*(1 + x + x^2 + 2*x^3 + 3*x^6 - x^7 - 2*x^8 - x^4 + x^9 + 4*x^10 + x^11) / ( (x - 1) * (x^4 + x^3 + x^2 + x + 1) * (x^8 - x^6 + x^4 - x^2 + 1) ). - R. J. Mathar, Jul 14 2012

Extensions

Added a(0)=0 from Vincenzo Librandi, Feb 04 2014