A143484 "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 number in position 1 of n-th permutation.

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

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.

Examples

			The full list of the 24 permutations is as follows (the present sequence gives the first column):
1 2 3 4
1 2 4 3
1 4 2 3
4 1 2 3
4 1 3 2
1 4 3 2
1 3 4 2
1 3 2 4
3 1 2 4
3 1 4 2
3 4 1 2
4 3 1 2
4 3 2 1
3 4 2 1
3 2 4 1
3 2 1 4
2 3 1 4
2 3 4 1
2 4 3 1
4 2 3 1
4 2 1 3
2 4 1 3
2 1 4 3
2 1 3 4
		

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 (2,-2,1,0,0,-1,2,-2,1,0,0,-1,2,-2,1,0,0,-1,2,-2,1).

Crossrefs

Cf. A143484-A143490, 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 bell(4)[modp(n-1,24)+1][1]: seq(a(n), n=1..121);

Mathematica

LinearRecurrence[{2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1}  {1, 1, 1, 4, 4, 1, 1, 1, 3, 3, 3, 4, 4, 3, 3, 3, 2, 2, 2, 4, 4}, 105] (* Jean-François Alcover, Mar 14 2021 *)

Formula

Period 24.