%I
%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 201205), 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 modulo5 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=vLhQfcZBWk">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(n1) + a(n2)  1) mod 5 + 1 for 3 <= n <= 18,
%F a(n) = (a(n1) + a(n2)  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
