A089911 a(n) = Fibonacci(n) mod 12.

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

Offset

0,4

Comments

From Reinhard Zumkeller, Jul 05 2013: (Start)

Sequence has been applied by several composers to 12-tone equal temperament pitch structure. The complete Fibonacci mod 12 system (a set of 10 periodic sequences) exhausts all possible ordered dyads; that is, every possible combination of two pitches is found in these sets.

a(A008594(n)) = 0;

a(A227144(n)) = 1;

a(3*A047522(n)) = 2;

a(A017569(n)) = a(2*A016933(n)) = a(4*A016777(n)) = 3;

a(2*A017629(n)) = a(3*A017137(n)) = a(6*A004767(n)) = 4;

a(A227146(n)) = 5;

a(nonexistent) = 6;

a(2*A017581(n)) = 7;

a(2*A017557(n)) = a(4*A016813(n)) = 8;

a(A017617(n)) = a(2*A016957(n)) = a(4*A016789(n)) = 9;

a(3*A047621(n)) = 10;

a(2*A017653(n)) = 11. (End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1199

Ron Knott, Fibonacci Numbers and the Golden Section

M. Renault, The Fibonacci Sequence Modulo M

A. P. Shah, Fibonacci Sequence Modulo m, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.

D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

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

Formula

Has period of 24, restricted period 12 and multiplier 5.

a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.

Maple

with(combinat, fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;

Mathematica

Table[Mod[Fibonacci[n], 12], {n, 0, 100}] (* Vincenzo Librandi, Feb 04 2014 *)

Prog

(Haskell)
a089911 n = a089911_list !! n
a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
                       (tail a089911_list) a089911_list
-- Reinhard Zumkeller, Jul 01 2013
(PARI) a(n)=fibonacci(n)%12 \\ Charles R Greathouse IV, Feb 03 2014
(Magma) [Fibonacci(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

Crossrefs

Cf. A000045, A001175, A003893, A010881.

Sequence in context: A093086 A093092 A031111 * A098978 A111301 A247193

Adjacent sequences: A089908 A089909 A089910 * A089912 A089913 A089914

Keyword

nonn,hear,easy

Author

Casey Mongoven, Nov 14 2003

Extensions

More terms from Ray Chandler, Nov 15 2003

Status

approved