OFFSET
1,2
COMMENTS
This is the "plain hunting" sequence with 4 bells.
a(n) is also the position of bell 4 (the tenor bell) in the (n+4)-th permutation of the "Fourth down, Extream between the two farthest Bells from it" bell-ringing permutation, A143484. - Alois P. Heinz, Aug 19 2008
Period 8 sequence: 1, 2, 3, 4, 4, 3, 2, 1, ... - Wesley Ivan Hurt, Mar 27 2014
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
R. Bailey, Change Ringing Resources
David Joyner, Application: Bell Ringing
M.I.T. Bell-Ringers, General Information On Change Ringing
Richard Duckworth and Fabian Stedman, Tintinnalogia, or, the Art of Ringing, (1671). Released by Project Gutenberg, 2006.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).
FORMULA
a(n) = (floor(-abs(n-(16*ceiling(n/8)-7)/2) + (16*ceiling(n/8)-7)/2)) mod 8. - Wesley Ivan Hurt, Mar 26 2014
G.f.: -x*(x^4+x^3+x^2+x+1) / ((x-1)*(x^4+1)). - Colin Barker, Mar 26 2014
EXAMPLE
The full list of the 24 permutations is as follows (the present sequence tells where the 1's are):
1 2 3 4
2 1 4 3
2 4 1 3
4 2 3 1
4 3 2 1
3 4 1 2
3 1 4 2
1 3 2 4
1 3 4 2
3 1 2 4
3 2 1 4
2 3 4 1
2 4 3 1
4 2 1 3
4 1 2 3
1 4 3 2
1 4 2 3
4 1 3 2
4 3 1 2
3 4 2 1
3 2 4 1
2 3 1 4
2 1 3 4
1 2 4 3
MAPLE
ring:= proc(k) option remember; local l, a, b, c, swap, h; l:= [1, 2, 3, 4]; swap:= proc(i, j) h:=l[i]; l[i]:=l[j]; l[j]:=h end; a:= proc() swap(1, 2); swap(3, 4); l[k] end; b:= proc() swap(2, 3); l[k] end; c:= proc() swap(3, 4); l[k] end; [l[k], seq([seq([a(), b()][], j=1..3), a(), c()][], i=1..3)] end: bells:=[seq(ring(k), k=1..4)]: indx:= proc(l, n, k) local i; for i from 1 to 4 do if l[i][n]=k then break fi od; i end: a:= n-> indx(bells, modp(n-1, 24)+1, 1): seq(a(n), n=1..99); # Alois P. Heinz, Aug 19 2008
MATHEMATICA
Table[Mod[Floor[-Abs[n-(16*Ceiling[n/8]-7)/2] + (16*Ceiling[n/8]-7)/2], 8], {n, 100}] (* Wesley Ivan Hurt, Mar 26 2014 *)
LinearRecurrence[{1, 0, 0, -1, 1}, {1, 2, 3, 4, 4}, 105] (* Jean-François Alcover, Mar 15 2021 *)
PROG
(Scheme) (define (A090281 n) (list-ref '(1 2 3 4 4 3 2 1) (modulo (- n 1) 8))) ;; Antti Karttunen, Aug 10 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 24 2004
STATUS
approved