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A082116
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Fibonacci sequence (mod 5).
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10
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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
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OFFSET
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0,4
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COMMENTS
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This sequence contains the complete set of residues modulo 5. See A079002. - Michel Marcus, Jan 31 2020
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REFERENCES
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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
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LINKS
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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.
Minjia Shi, 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).
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FORMULA
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Sequence is periodic with Pisano period 20.
a(n) = 1/380*{ - 15*(n mod 20) + 23*[(n + 1) mod 20] + 61*[(n + 2) mod 20] - 34*[(n + 3) mod 20] + 4*[(n + 4) mod 20] - 34*[(n + 5) mod 20] + 42*[(n + 6) mod 20] + 23*[(n + 7) mod 20] + 23*[(n + 8) mod 20] + 4*[(n + 9) mod 20] - 72*[(n + 10) mod 20] + 80*[(n + 11) mod 20] - 53*[(n + 12) mod 20] + 42*[(n + 13) mod 20] + 4*[(n + 14) mod 20] - 53*[(n + 15) mod 20] + 61*[(n + 16) mod 20] - 15*[(n + 17) mod 20] - 15*[(n + 18) mod 20] + 4*[(n + 19) mod 20]} with n> = 0. - Paolo P. Lava, Dec 20 2006
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
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MATHEMATICA
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Table[Mod[Fibonacci[n], 5], {n, 0, 125}] (* Alonso del Arte, Jul 29 2013 *)
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PROG
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(MAGMA) [Fibonacci(n) mod 5: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014
(PARI) a(n)=fibonacci(n)%5 \\ Charles R Greathouse IV, Oct 07 2015
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CROSSREFS
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Cf. A000045, A011655, A082115, A079343, A082116, A082117, A079344, A079002.
Sequence in context: A051933 A234963 A131900 * A079777 A224909 A227536
Adjacent sequences: A082113 A082114 A082115 * A082117 A082118 A082119
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KEYWORD
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nonn,easy
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AUTHOR
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Eric W. Weisstein, Apr 03 2003
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EXTENSIONS
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Added a(0)=0 from Vincenzo Librandi, Feb 04 2014
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STATUS
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approved
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