A082117 -id:A082117 - cp's OEIS frontend

A010074 a(n) = sum of base-7 digits of a(n-1) + sum of base-7 digits of a(n-2).

0, 1, 1, 2, 3, 5, 8, 7, 3, 4, 7, 5, 6, 11, 11, 10, 9, 7, 4, 5, 9, 8, 5, 7, 6, 7, 7, 2, 3, 5, 8, 7, 3, 4, 7, 5, 6, 11, 11, 10, 9, 7, 4, 5, 9, 8, 5, 7, 6, 7, 7, 2, 3, 5, 8, 7, 3, 4, 7, 5, 6, 11, 11, 10, 9, 7, 4, 5, 9, 8, 5, 7, 6, 7, 7

Offset: 0

N. J. A. Sloane, Leonid Broukhis

The digital sum analog (in base 7) of the Fibonacci recurrence. - Hieronymus Fischer, Jun 27 2007
a(n) and Fib(n)=A000045(n) are congruent modulo 6 which implies that (a(n) mod 6) is equal to (Fib(n) mod 6) = A082117(n-1) (for n>0). Thus (a(n) mod 6) is periodic with the Pisano period A001175(6)=24. - Hieronymus Fischer, Jun 27 2007
For general bases p>2, the inequality 2<=a(n)<=2p-3 holds (for n>2). Actually, a(n)<=11=A131319(7) for the base p=7. - Hieronymus Fischer, Jun 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for Colombian or self numbers and related sequences

Cf. A000045, A010073, A010075, A010076, A010077, A131294, A131295, A131296, A131297, A131318, A131319, A131320.

nxt[{a_,b_}]:={b,Total[IntegerDigits[a,7]]+Total[IntegerDigits[b,7]]}; Transpose[NestList[nxt,{0,1},80]][[1]] (* Harvey P. Dale, Oct 12 2013 *)

Periodic from n=3 with period 24. - Franklin T. Adams-Watters, Mar 13 2006
From Hieronymus Fischer, Jun 27 2007: (Start)
a(n) = a(n-1)+a(n-2)-6*(floor(a(n-1)/7)+floor(a(n-2)/7)).
a(n) = floor(a(n-1)/7)+floor(a(n-2)/7)+(a(n-1)mod 7)+(a(n-2)mod 7).
a(n) = (a(n-1)+a(n-2)+6*(A010876(a(n-1))+A010876(a(n-2))))/7.
a(n) = Fib(n)-6*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

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

A280154 a(n) = 5*Lucas(n).

Original entry on oeis.org

10, 5, 15, 20, 35, 55, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6820, 11035, 17855, 28890, 46745, 75635, 122380, 198015, 320395, 518410, 838805, 1357215, 2196020, 3553235, 5749255, 9302490, 15051745, 24354235, 39405980, 63760215, 103166195, 166926410, 270092605, 437019015

Offset: 0

Bruno Berselli, Dec 27 2016

Fibonacci sequence beginning 10, 5.
After 5, the sequence provides the 3rd column of the rectangular array in A213590.
After 5, all terms belong to A191921 because a(n) = Lucas(n+4) - 3*Lucas(n-1).
From G. C. Greubel, Dec 27 2016: (Start)
{a(n) mod 3} yields (1,2,0,2,2,1,0,1), repeated, and is given as A082115.
{a(n) mod 6} yields (4,5,3,2,5,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1) and is given as A082117. (End)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A084176.
Cf. A022088: 5*Fibonacci(n).
Cf. A022359: Lucas(n+5) + Lucas(n-5).
Cf. A000032, A000045, A191921, A213590.
Cf. sequences with formula Fibonacci(n+k) + Fibonacci(n-k): A006355 (k=0, without the initial 1), A000032 (k=1), A022086 (k=2), A022112 (k=3, with an initial 4), A022090 (k=4), this sequence (k=5), A022352 (k=6).

Magma
```
[5*Lucas(n): n in [0..40]];
```

F := n -> combinat:-fibonacci(n):
seq(F(n+5) + F(n-5), n=0..38); # Peter Luschny, Dec 29 2016

Mathematica
```
Table[5 LucasL[n], {n, 0, 40}]
```

vector(40, n, n--; fibonacci(n+5)+fibonacci(n-5))

def A280154():
    x, y = 10, 5
    while True:
        yield x
        x, y = y, x + y
a = A280154(); print([next(a) for  in range(39)]) # _Peter Luschny, Dec 29 2016

G.f.: 5*(2 - x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2) for n>1.
a(n) = Fibonacci(n+5) + Fibonacci(n-5), with Fibonacci(-k) = -(-1)^k*Fibonacci(k) for the negative indices.

A004699 a(n) = floor(Fibonacci(n)/6).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 9, 14, 24, 38, 62, 101, 164, 266, 430, 696, 1127, 1824, 2951, 4776, 7728, 12504, 20232, 32736, 52968, 85704, 138673, 224378, 363051, 587429, 950481, 1537910, 2488392, 4026302, 6514694, 10540997, 17055692, 27596690, 44652382

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1).

Cf. A000045.

[Floor(Fibonacci(n)/6): n in [0..40]]; // Vincenzo Librandi, Jul 10 2012

seq(floor(combinat[fibonacci](n)/6), n=0..40); # Muniru A Asiru, Oct 10 2018

Table[Floor[Fibonacci[n]/6], {n, 0, 50}] (* Vincenzo Librandi, Jul 10 2012 *)
CoefficientList[Series[x^6 (1 + x + x^4 + x^6 + x^9 + x^10 + x^11 + x^14 + x^15 + x^17 + x^18)/((1 - x - x^2) (1 - x^24)), {x, 0, 50}], x] (* Stefano Spezia, Oct 11 2018 - corrected by G. C. Greubel, May 21 2019 *)

vector(50, n, n--; fibonacci(n)\6) \\ G. C. Greubel, Oct 09 2018

[floor(fibonacci(n)/6) for n in (0..40)] # G. C. Greubel, May 21 2019

G.f.: x^6*(1 + x + x^4 + x^6 + x^9 + x^10 + x^11 + x^14 + x^15 + x^17 + x^18)/((1 - x - x^2)*(1 - x^24)). [Corrected by G. C. Greubel, May 21 2019]

a(n) = (A000045(n) - A082117(n))/6. - R. J. Mathar, Jul 14 2012

A269701 Cyclic Fibonacci sequence, restricted to maximum=6.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 6, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 6, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 6, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 6, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 6, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 6, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 6, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 6, 1

Offset: 0

Gabriel Osorio, Mar 04 2016

Sequence has a period of 24.

			For n = 6; F(5) + F(4) equals 8 then F(6) = 8 - 6 = 2.

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. A000045 (Fibonacci numbers), A082117.

fibocy(1) -> 1;
fibocy(2) -> 1;
fibocy(N) when N > 1 ->
   Tmp = fibocy(N-1) + fibocy(N-2),
   if Tmp > 6 -> Tmp - 6;
      true -> Tmp
   end.

A269701 := proc(n)
    option remember;
    if n <=5 then
        combinat[fibonacci](n) ;
    else
        a := procname(n-1)+procname(n-2) ;
        if a > 6 then
            a-6;
        else
            a;
        end if;
    end if;
end proc: # R. J. Mathar, Apr 16 2016

Table[Mod[Fibonacci[n], 6], {n, 100}] /. 0 -> 6 (* Alonso del Arte, Mar 28 2016 *)
PadRight[{0},120,{6,1,1,2,3,5,2,1,3,4,1,5,6,5,5,4,3,1,4,5,3,2,5,1}] (* Harvey P. Dale, May 13 2019 *)

F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1 and F(n) = F(n-1) + F(n-2) - 6 if F(n-1) + F(n-2) > 6.

G.f.: ( -x *(1 +x +2*x^2 +3*x^3 +5*x^4 +2*x^5 +x^6 +3*x^7 +4*x^8 +x^9 +5*x^10 +6*x^11 +5*x^12 +5*x^13 +4*x^14 +3*x^15 +x^16 +4*x^17 +5*x^18 +3*x^19 +2*x^20 +5*x^21 +x^22 +6*x^23) ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Apr 16 2016

A160101 Lodumo_6 of Fibonacci numbers.

Original entry on oeis.org

0, 1, 7, 2, 3, 5, 8, 13, 9, 4, 19, 11, 6, 17, 23, 10, 15, 25, 16, 29, 21, 14, 35, 31, 12, 37, 43, 20, 27, 41, 26, 49, 33, 22, 55, 47, 18, 53, 59, 28, 39, 61, 34, 65, 45, 32, 71, 67, 24, 73, 79, 38, 51, 77, 44, 85, 57, 40, 91, 83, 30, 89, 95, 46, 63, 97, 52, 101, 69, 50, 107, 103

Offset: 0

Philippe Deléham, May 01 2009

Permutation of nonnegative integers.

Cf. A000045, A082117

a(n)=lod_6(A000045(n)).

Corrected by R. J. Mathar, May 02 2009

A160187 Lodumo_6 of Lucas numbers.

Original entry on oeis.org

2, 1, 3, 4, 7, 5, 0, 11, 17, 10, 9, 13, 16, 23, 15, 8, 29, 19, 6, 25, 31, 14, 21, 35, 20, 37, 27, 22, 43, 41, 12, 47, 53, 28, 33, 49, 34, 59, 39, 26, 65, 55, 18, 61, 67, 32, 45, 71, 38, 73, 51, 40, 79, 77, 24, 83, 89, 46, 57, 85, 52, 95, 63, 44, 101, 91, 30, 97, 103, 50, 69, 107

Offset: 0

Philippe Deléham, May 03 2009

Permutation of nonnegative integers.

Cf. A000032, A082117

a(n)=lod_6(A000032(n)).

cp's OEIS Frontend

A010074 a(n) = sum of base-7 digits of a(n-1) + sum of base-7 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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A079344 F(n) mod 8, where F(n) = A000045(n) is the n-th Fibonacci number.

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A280154 a(n) = 5*Lucas(n).

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A004699 a(n) = floor(Fibonacci(n)/6).

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A269701 Cyclic Fibonacci sequence, restricted to maximum=6.