A143490 "Fourth down, Extream [sic] between the two farthest Bells from it" in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(1,2,4,3), .. which runs through all permutations of {1,2,3,4} with period 24; sequence gives position of bell 3 in n-th permutation.

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

Offset: 1

Author

Alois P. Heinz, Aug 19 2008

Keywords

• nonn
• easy

Comments

Start with (1,2,3,4), i.e. the first permutation of {1,2,3} followed by 4; then for each next permutation, transpose 4 one to the left; if at position 1, replace {1,2,3} recursively by the next permutation of these numbers. Thereafter, for each next permutation, transpose 4 to the right. And so on.

Links

• Richard Duckworth and Fabian Stedman, Tintinnalogia, or, the Art of Ringing, (1671). Released by Project Gutenberg, 2006.
• Index entries for sequences related to bell ringing
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,-1,1).

Crossrefs

Cf. A143484, A143485, A143486, A143487, A143488, A143489, A090281.

Programs

Maple

ring:= proc(k::nonnegint) local p,i,left,l,nf,ini; if k<=1 then proc() [1$k] end else ini := proc() p:= ring(k-1); i:= k; left:= true; l:= p(); nf:= k! end; ini(); proc() local ll; ll:= [seq(l[t], t=1..(i-1)), k, seq(l[t], t=i..(k-1))]; if left then if i>1 then i:= i-1 else left:= false; l:=p() fi else if i indx (bell(4)[modp(n-1,24)+1], 3): seq (a(n), n=1..121);

Mathematica

a[n_] := a[n] = If[n <= 13, {3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 1, 2, 2}[[n]], a[n-1] - a[n-12] + a[n-13]]; Array[a, 105] (* Jean-François Alcover, May 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,-1,1},{3,4,4,4,3,3,2,2,1,1,1,2,2},120] (* Harvey P. Dale, Apr 28 2020 *)

Formula

Period 24.