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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 Richard Duckworth and Fabian Stedman, Tintinnalogia, or, the Art of Ringing, Project Gutenberg. 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 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.)