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A143486
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"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 3 of n-th permutation.
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6
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3, 4, 2, 2, 3, 3, 4, 2, 2, 4, 1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 1, 1, 4, 3, 3, 4, 2, 2, 3, 3, 4, 2, 2, 4, 1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 1, 1, 4, 3, 3, 4, 2, 2, 3, 3, 4, 2, 2, 4, 1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 1, 1, 4, 3, 3, 4, 2, 2, 3, 3, 4, 2, 2, 4, 1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 1, 1, 4, 3, 3, 4, 2, 2, 3, 3, 4, 2, 2
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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Period 24.
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) - a(n-6) + 2*a(n-7) - 2*a(n-8) + a(n-9) - a(n-12) + 2*a(n-13) - 2*a(n-14) + a(n-15) - a(n-18) + 2*a(n-19) - 2*a(n-20) + a(n-21) for n > 21.
G.f.: x*(-3*x^20 + 2*x^19 + x^18 - 4*x^17 + x^16 + 2*x^15 - 6*x^14 + 6*x^13 - 3*x^12 - 4*x^11 + 6*x^10 - 3*x^9 - 3*x^8 + 5*x^7 - 5*x^6 + x^5 + x^4 - 3*x^3 + 2*x - 3)/((x - 1)*(x^2 + 1)*(x^4 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^8 - x^4 + 1)). (End)
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MAPLE
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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][3]: seq (a(n), n=1..121);
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MATHEMATICA
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LinearRecurrence[{2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 2, -2, 1}, {3, 4, 2, 2, 3, 3, 4, 2, 2, 4, 1, 1, 2, 2, 4, 1, 1, 4, 3, 3, 1}, 105] (* Jean-François Alcover, Mar 14 2021 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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