A105870 Fibonacci sequence (mod 7).

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

Offset

0,4

Comments

Sequence is periodic with Pisano period 16 = A001175(7).

Links

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

Brady Haran, Fibonacci Tartan and Bagpipes, Numberphile video (2013). The music by Alan Stewart at 1:53 to 3:20 has pitch based on this sequence.

Wayne Peng, ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields, arXiv:1511.05645 [math.NT], 2015.

Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 10.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Formula

G.f.: - x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + x^5 + 6*x^6 + 6*x^8 + 6*x^9 + 5*x^10 + 4*x^11 + 2*x^12 + 6*x^13 + x^14)/((x - 1)*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)). - R. J. Mathar, Jul 14 2012

a(1) = a(2) = 1, then a(n) = (a(n - 2) + a(n - 1)) mod 7. - Alonso del Arte, Jul 30 2013

Example

a(5) = 5 because Fibonacci(5) = 5.
a(6) = 1 because Fibonacci(6) = 8 and 8 mod 7 = 1.
a(7) = 6 because Fibonacci(7) = 13 and 13 mod 7 = 6.

Mathematica

Table[Mod[Fibonacci[n], 7], {n, 0, 100}] (* Alonso del Arte, Jul 29 2013 *)

Prog

(PARI) a(n)=fibonacci(n)%7 \\ Charles R Greathouse IV, Jun 04 2013
(PARI) a(n)=lift(((Mod([1, 1; 1, 0], 7))^n)[1, 2]) \\ Charles R Greathouse IV, Jun 04 2013
(PARI) a(n)=fibonacci(n%16)%7 \\ Charles R Greathouse IV, Jan 06 2016
(Haskell)
a105870 n = a105870_list !! (n-1)
a105870_list = 1 : 1 : zipWith (\u v -> (u + v) `mod` 7)
                               (tail a105870_list) a105870_list
-- Reinhard Zumkeller, Jan 15 2014
(Magma) [Fibonacci(n) mod 7: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014
(Python)
A105870_list, a, b, = [], 0, 1
for _ in range(10**3):
    A105870_list.append(a)
    a, b = b, (a+b) % 7 # Chai Wah Wu, Nov 26 2015

Crossrefs

Sequence in context: A239693 A256655 A128047 * A328145 A096534 A322558

Adjacent sequences: A105867 A105868 A105869 * A105871 A105872 A105873

Keyword

nonn,easy,hear

Author

Shyam Sunder Gupta, May 05 2005

Extensions

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

Status

approved