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A143488 "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 1 (the treble bell) in n-th permutation. 4
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, 1, 1, 2, 2, 1, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
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, Project Gutenberg.

Index entries for sequences related to bell ringing

FORMULA

Period 24.

From Chai Wah Wu, Jan 15 2020: (Start)

a(n) = a(n-1) - a(n-12) + a(n-13) for n > 13.

G.f.: x*(-2*x^12 - x^10 - x^8 + x^5 - x^3 - 1)/(x^13 - x^12 + x - 1). (End)

EXAMPLE

The full list of the 24 permutations is as follows (the present sequence gives position of bell 1):

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: 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], 1): seq(a(n), n=1..121);

MATHEMATICA

LinearRecurrence[

     {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1},

     {1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 4, 4, 4, 3}, 105]

(* Jean-François Alcover, Mar 15 2021 *)

CROSSREFS

Cf. A143484, A143485, A143486, A143487, A143489, A143490, A090281.

Sequence in context: A238013 A303940 A280134 * A201159 A047070 A071127

Adjacent sequences:  A143485 A143486 A143487 * A143489 A143490 A143491

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 19 2008

STATUS

approved

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Last modified July 28 13:15 EDT 2021. Contains 346332 sequences. (Running on oeis4.)