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

Author

Casey Mongoven, Nov 14 2003

Keywords

• nonn
• hear
• easy

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).

Crossrefs

Cf. A000045, A001175, A003893, A010881.

Programs

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

Magma

[Fibonacci(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

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 *)

PARI

a(n)=fibonacci(n)%12 \\ Charles R Greathouse IV, Feb 03 2014

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.

Extensions

More terms from Ray Chandler, Nov 15 2003