A079344 -id:A079344 - cp's OEIS frontend

A082118 Duplicate of A079344.

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

Offset: 0

dead

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

A010076 a(n) = sum of base-9 digits of a(n-1) + sum of base-9 digits of a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9, 8, 9, 9

Offset: 0

N. J. A. Sloane, Leonid Broukhis

The digital sum analog (in base 9) of the Fibonacci recurrence. - Hieronymus Fischer, Jun 27 2007
a(n) and Fib(n)=A000045(n) are congruent modulo 8 which implies that (a(n) mod 8) is equal to (Fib(n) mod 8) = A079344(n). Thus (a(n) mod 8) is periodic with the Pisano period A001175(8)=12. - Hieronymus Fischer, Jun 27 2007
For general bases p>2, we have the inequality 2<=a(n)<=2p-3 (for n>2). Actually, a(n)<=13=A131319(9) for the base p=9. - Hieronymus Fischer, Jun 27 2007

Index entries for Colombian or self numbers and related sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045, A001175, A010073, A010074, A010075, A010077, A010878, A079344, A131294, A131295, A131296, A131297, A131318, A131319, A131320.

PadRight[{0, 1, 1}, 100, {8, 9, 9, 2, 3, 5, 8, 13, 13, 10, 7, 9}] (* Paolo Xausa, Aug 25 2024 *)

Periodic from n=3 with period 12. - Franklin T. Adams-Watters, Mar 13 2006
From Hieronymus Fischer, Jun 27 2007: (Start)
a(n) = a(n-1)+a(n-2)-8*(floor(a(n-1)/9)+floor(a(n-2)/9)).
a(n) = floor(a(n-1)/9)+floor(a(n-2)/9)+(a(n-1)mod 9)+(a(n-2)mod 9).
a(n) = (a(n-1)+a(n-2)+8*(A010878(a(n-1))+A010878(a(n-2))))/9.
a(n) = Fib(n)-8*sum{1A000045(n). (End)

Incorrect comment removed by Michel Marcus, Apr 29 2018

A082115 Fibonacci sequence (mod 3).

Original entry on oeis.org

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

Eric W. Weisstein, Apr 03 2003

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).

Cf. A011655, A079343, A082116, A082117, A105870, A079344.

[Fibonacci(n) mod 3: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

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 *)

a(n)=fibonacci(n%8)%3 \\ Charles R Greathouse IV, Sep 28 2015

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

Added a(0)=0. - Jon Perry, Sep 15 2013

A082116 Fibonacci sequence (mod 5).

Original entry on oeis.org

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

Eric W. Weisstein, Apr 03 2003

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

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

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).

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

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

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

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

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

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

A082117 Fibonacci sequence (mod 6).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0, 1, 1, 2, 3, 5, 2

Offset: 0

Eric W. Weisstein, Apr 03 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Syamsudhuha Muslim, Sri Gemawati, The Identity of Fibonacci Numbers Z6, International Mathematical Forum, Vol. 13, 2018, no. 5, 225-231.
Eric Weisstein's World of Mathematics, Fibonacci Number
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, 1).

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

[Fibonacci(n) mod 6: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

Table[Mod[Fibonacci[n], 6], {n, 0, 100}] (* Alonso del Arte, Jul 29 2013 *)

a(n)=fibonacci(n)%6 \\ Charles R Greathouse IV, Oct 07 2016

Sequence is periodic with Pisano period 24 = A001175(6).

G.f.: -x*(x^22 + 5*x^21 + 2*x^20 + 3*x^19 + 5*x^18 + 4*x^17 + x^16 + 3*x^15 + 4*x^14 + 5*x^13 + 5*x^12 + 5*x^10 + x^9 + 4*x^8 + 3*x^7 + x^6 + 2*x^5 + 5*x^4 + 3*x^3 + 2*x^2 + x + 1)/((x - 1)*(x + 1)*(x^2 - x + 1)*(x^2 + 1)*(x^2 + x + 1)*(x^4 - x^2 + 1)*(x^4 + 1)*(x^8 - x^4 + 1)). - Colin Barker, Aug 15 2012

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

A111958 Lucas numbers (A000032) mod 8.

Original entry on oeis.org

2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7

Offset: 0

N. J. A. Sloane, Nov 28 2005

nonn

This sequence has period-length 12.

G. C. Greubel, Table of n, a(n) for n = 0..9999
Paulo Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Quart. 43 (No. 1, 2005), 3-14.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1).

Cf. A000032, A079344, A130893.

LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}, {2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3}, 105] (* Ray Chandler, Aug 27 2015 *)
Mod[LucasL[Range[0, 99]], 8] (* Alonso del Arte, Dec 19 2015 *)

From G. C. Greubel, Feb 08 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11).
a(n+12) = a(n). (End)

A079345 Fibonacci(n) mod 16.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 5, 2, 7, 9, 0, 9, 9, 2, 11, 13, 8, 5, 13, 2, 15, 1, 0, 1, 1, 2, 3, 5, 8, 13, 5, 2, 7, 9, 0, 9, 9, 2, 11, 13, 8, 5, 13, 2, 15, 1, 0, 1, 1, 2, 3, 5, 8, 13, 5, 2, 7, 9, 0, 9, 9, 2, 11, 13, 8, 5, 13, 2, 15, 1, 0, 1, 1, 2, 3, 5, 8, 13, 5, 2, 7, 9, 0, 9, 9, 2, 11, 13, 8, 5, 13, 2

Offset: 0

Jon Perry, Jan 04 2003

Periodic with period 24. - Jon Perry, Jan 08 2003

			a(8) = F(8) mod 16 = 21 mod 16 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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, 1).

Cf. A079343, A079344.

[Fibonacci(n) mod 16: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

a={};Do[f=Fibonacci[n];AppendTo[a,Mod[f,16]],{n,1,30}];a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
Table[Mod[Fibonacci[n], 16], {n, 0, 100}] (* Vincenzo Librandi, Feb 04 2014 *)

for (n=1,100,print1(fibonacci(n)%16","))

A160108 Lodumo_8 of Fibonacci numbers.

Original entry on oeis.org

0, 1, 9, 2, 3, 5, 8, 13, 21, 10, 7, 17, 16, 25, 33, 18, 11, 29, 24, 37, 45, 26, 15, 41, 32, 49, 57, 34, 19, 53, 40, 61, 69, 42, 23, 65, 48, 73, 81, 50, 27, 77, 56, 85, 93, 58, 31, 89, 64, 97, 105, 66, 35, 101, 72, 109, 117, 74, 39, 113, 80, 121, 129, 82, 43, 125, 88, 133, 141

Offset: 0

Philippe Deléham, May 02 2009

nonn

Some integers (see A047406) are missing.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000045, A079344.

Lodumo_8 transform of A000045 (for definition see Transforms).
Conjecture: a(n) = 2*a(n-6)-a(n-12). - Colin Barker, Oct 04 2014
Empirical g.f.: x*(7*x^10 +x^9 +6*x^8 +3*x^7 +11*x^6 +8*x^5 +5*x^4 +3*x^3 +2*x^2 +9*x +1) / ((x -1)^2*(x +1)^2*(x^2 -x +1)^2*(x^2 +x +1)^2). - Colin Barker, Oct 04 2014

Replaced 35 by 33 - R. J. Mathar, May 03 2009

cp's OEIS Frontend

A082118 Duplicate of A079344.

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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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A010076 a(n) = sum of base-9 digits of a(n-1) + sum of base-9 digits of a(n-2).

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A082115 Fibonacci sequence (mod 3).

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A082116 Fibonacci sequence (mod 5).

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A082117 Fibonacci sequence (mod 6).

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A111958 Lucas numbers (A000032) mod 8.

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A079345 Fibonacci(n) mod 16.

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