A090281 "Plain Bob Minimus" in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(2,1,4,3), ... which runs through all permutations of {1,2,3,4} with period 24; sequence gives position of bell 1 (the treble bell) in n-th permutation.
1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3
Offset: 1
Examples
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
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).
- Index entries for sequences related to bell ringing
Programs
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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
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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 *) -
Scheme
(define (A090281 n) (list-ref '(1 2 3 4 4 3 2 1) (modulo (- n 1) 8))) ;; Antti Karttunen, Aug 10 2017
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
Comments