A172984 - OEIS

%I #27 Feb 24 2019 02:08:32

%S 1,1,2,3,5,3,3,1,4,5,4,4,3,2,5,2,2,4,2

%N For n <= 18, a(n) = Fibonacci(n) mod 5, using representatives {5,1,2,3,4} (i.e., 5 instead of the usual 0), and a(19)=2.

%C Previous name was: Sequence congruent to the Fibonacci sequence modulo 5, with 1 added to the last term. (Seen on "Mathnet.")

%C This sequence was used as a puzzle in the "Mathnet" portion of the children's mathematics television show Square One TV. In the series "Case of the Willing Parrot" (Episodes 201-205), the sequence was found in a pattern of tiles on a wall (see Example). The mystery was solved through the identification of the sequence, whose terms (with the exception of the last) were discovered to follow a pattern: after the two initial 1's, each term was the sum of the two previous terms, with 5 subtracted from any sum that exceeded 5. Thus, with the exception of the final term, the terms were the modulo-5 residues of the Fibonacci sequence (with a 5 representing each term that would have been a 0); see Formula section. After the anomalous final term was recognized as a clue, the physical removal of the upper tile of the rightmost column, which represented that final term, revealed a hidden key to a safe deposit box.

%D Schneider, Joel, et al., Square One TV: Season Two Content Analysis and Show Rundowns. Jul 21 1988, Children's Television Workshop; New York.

%H Children's Television Workshop, <a href="https://www.youtube.com/watch?v=vLhQfcZ-BWk">Mathnet - Case of the Willing Parrot (Recap & Finale) Pt. 2</a> (video)

%H Schneider, Joel, et al., <a href="http://www.squareonetv.org/default.asp?ID=17">Mathnet Guide</a>

%F Recursive formula:

%F   a(1) = 1, a(2) = 1,

%F   a(n) = (a(n-1) + a(n-2) - 1) mod 5 + 1 for 3 <= n <= 18,

%F   a(n) = (a(n-1) + a(n-2) - 1) mod 5 + 2 for n = 19.

%F Result expressed in terms of Fibonacci sequence:

%F   a(n) = (Fibonacci(n) - 1) mod 5 + 1 for 1 <= n <= 18;

%F   a(n) = (Fibonacci(n) - 1) mod 5 + 2 for n = 19.

%e : The following pattern of tiles is shown in the video:

%e :

%e :          X         X         X

%e :          X       X X X X     X     X

%e :        X X X X   X X X X X   X     X

%e :      X X X X X   X X X X X X X X X X X

%e :  X X X X X X X X X X X X X X X X X X X

%o (PARI) a(n)=if(n>19,return(0)); if(n==19,return(2)); my(t=fibonacci(n)%5);if(t==0,t=5);return(t); \\ _Joerg Arndt_, Dec 01 2014

%Y Cf. A000045, A082116.

%K nonn,fini,full

%O 1,3

%A Jon Suen (jsuen(AT)ece.ucsb.edu), Feb 06 2010

%E Edited by _Jon E. Schoenfield_, Dec 01 2014

%E Edited and new name from _Joerg Arndt_, Dec 01 2014