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

%I A089911 #60 Jan 05 2025 19:51:37
%S A089911 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,
%T A089911 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,
%U A089911 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
%N A089911 a(n) = Fibonacci(n) mod 12.
%C A089911 From _Reinhard Zumkeller_, Jul 05 2013: (Start)
%C A089911 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.
%C A089911 a(A008594(n)) = 0;
%C A089911 a(A227144(n)) = 1;
%C A089911 a(3*A047522(n)) = 2;
%C A089911 a(A017569(n)) = a(2*A016933(n)) = a(4*A016777(n)) = 3;
%C A089911 a(2*A017629(n)) = a(3*A017137(n)) = a(6*A004767(n)) = 4;
%C A089911 a(A227146(n)) = 5;
%C A089911 a(nonexistent) = 6;
%C A089911 a(2*A017581(n)) = 7;
%C A089911 a(2*A017557(n)) = a(4*A016813(n)) = 8;
%C A089911 a(A017617(n)) = a(2*A016957(n)) = a(4*A016789(n)) = 9;
%C A089911 a(3*A047621(n)) = 10;
%C A089911 a(2*A017653(n)) = 11. (End)
%H A089911 Reinhard Zumkeller, <a href="/A089911/b089911.txt">Table of n, a(n) for n = 0..1199</a>
%H A089911 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html">Fibonacci Numbers and the Golden Section</a>
%H A089911 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">The Fibonacci Sequence Modulo M</a>
%H A089911 A. P. Shah, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-2/shah.pdf">Fibonacci Sequence Modulo m</a>, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.
%H A089911 D. D. Wall, <a href="http://www.jstor.org/stable/2309169">Fibonacci series modulo m</a>, Amer. Math. Monthly, 67 (1960), 525-532.
%H A089911 <a href="/index/Rec#order_22">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1).
%F A089911 Has period of 24, restricted period 12 and multiplier 5.
%F A089911 a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.
%p A089911 with(combinat,fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;
%t A089911 Table[Mod[Fibonacci[n], 12], {n, 0, 100}] (* _Vincenzo Librandi_, Feb 04 2014 *)
%o A089911 (Haskell)
%o A089911 a089911 n = a089911_list !! n
%o A089911 a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
%o A089911                        (tail a089911_list) a089911_list
%o A089911 -- _Reinhard Zumkeller_, Jul 01 2013
%o A089911 (PARI) a(n)=fibonacci(n)%12 \\ _Charles R Greathouse IV_, Feb 03 2014
%o A089911 (Magma) [Fibonacci(n) mod 12: n in [0..100]]; // _Vincenzo Librandi_, Feb 04 2014
%Y A089911 Cf. A000045, A001175, A003893, A010881.
%K A089911 nonn,hear,easy
%O A089911 0,4
%A A089911 _Casey Mongoven_, Nov 14 2003
%E A089911 More terms from _Ray Chandler_, Nov 15 2003