A172984 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.

1, 1, 2, 3, 5, 3, 3, 1, 4, 5, 4, 4, 3, 2, 5, 2, 2, 4, 2

Offset

1,3

Comments

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

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.

References

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

Links

Table of n, a(n) for n=1..19.

Children's Television Workshop, Mathnet - Case of the Willing Parrot (Recap & Finale) Pt. 2 (video)

Schneider, Joel, et al., Mathnet Guide

Formula

Recursive formula:

a(1) = 1, a(2) = 1,

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

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

Result expressed in terms of Fibonacci sequence:

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

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

Example

: The following pattern of tiles is shown in the video:
:
:          X         X         X
:          X       X X X X     X     X
:        X X X X   X X X X X   X     X
:      X X X X X   X X X X X X X X X X X
:  X X X X X X X X X X X X X X X X X X X

Prog

(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

Crossrefs

Cf. A000045, A082116.

Sequence in context: A322235 A346477 A337583 * A072751 A251542 A131971

Adjacent sequences: A172981 A172982 A172983 * A172985 A172986 A172987

Keyword

nonn,fini,full

Author

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

Extensions

Edited by Jon E. Schoenfield, Dec 01 2014

Edited and new name from Joerg Arndt, Dec 01 2014

Status

approved