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

1,4

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

Table of n, a(n) for n=1..105.

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).

Formula

Period 24.

Example

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

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<k then i:= i+1 else left:= true; l:=p() fi fi; nf:= nf-1; if nf = 0 then ini() fi; ll end fi end: bell := proc(k) option remember; local p; p:= ring(k); [seq(p(), i=1..k!)] end: a := n-> 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 *)

Crossrefs

Cf. A143484-A143490, A090281.

Sequence in context: A046553 A046552 A155454 * A047214 A307886 A351122

Adjacent sequences: A143481 A143482 A143483 * A143485 A143486 A143487

Keyword

nonn,easy

Author

Alois P. Heinz, Aug 19 2008

Status

approved