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

1,1

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

Formula

Period 24.

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: indx:= proc(l, k) local i; for i from 1 to nops(l) do if l[i]=k then break fi od; i end: a := n-> 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 *)

Crossrefs

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

Sequence in context: A279781 A262827 A347827 * A007485 A280356 A018244

Adjacent sequences: A143487 A143488 A143489 * A143491 A143492 A143493

Keyword

nonn,easy

Author

Alois P. Heinz, Aug 19 2008

Status

approved