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%I #24 Mar 18 2017 10:51:58
%S 3,4,1,3,2,1,4,2,4,2,1,4,3,1,2,3,2,3,1,2,4,1,3,4,3,4,1,3,2,1,4,2,4,2,
%T 1,4,3,1,2,3,2,3,1,2,4,1,3,4,3,4,1,3,2,1,4,2,4,2,1,4,3,1,2,3,2,3,1,2,
%U 4,1,3,4,3,4,1,3,2,1,4,2,4,2,1,4,3,1,2,3,2,3,1,2,4,1,3,4,3,4,1
%N "Plain Bob Minimus" in bell-ringing is a sequence of permutations p_1=(1,2,3,4), p_2=(2,1,4,3), .. which runs through all permutations of {1,2,3,4} with period 24; sequence gives number in position 3 of n-th permutation.
%H R. Bailey, <a href="http://www.ringing.info">Change Ringing Resources</a>
%H David Joyner, <a href="http://www.usna.edu/Users/math/wdj/book/node158.html">Application: Bell Ringing</a>
%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1).
%H <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>
%F Period 24.
%F From _Chai Wah Wu_, Jul 17 2016: (Start)
%F a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) - a(n-14) + a(n-15) - a(n-16) + a(n-17) - a(n-18) + a(n-19) - a(n-20) + a(n-21) - a(n-22) + a(n-23) for n > 23.
%F G.f.: x*(-4*x^22 + x^21 - 2*x^20 - 2*x^19 - x^17 - 2*x^16 - 3*x^14 + x^13 - 2*x^12 - x^11 - 3*x^10 + 2*x^9 - 4*x^8 - 2*x^6 - 2*x^5 + x^4 - 3*x^3 - x - 3)/((x - 1)*(x^2 + 1)*(x^4 + 1)*(x^2 - x + 1)*(x^2 + x + 1)*(x^4 - x^2 + 1)*(x^8 - x^4 + 1)). (End)
%p ring:= proc(k) option remember; local l, a, b, c, swap, h; l:= [1,2,3,4]; swap:= proc(i,j) h:=l[i]; l[i]:=l[j]; l[j]:=h end; a:= proc() swap(1,2); swap(3,4); l[k] end; b:= proc() swap(2,3); l[k] end; c:= proc() swap(3,4); l[k] end; [l[k], seq([seq([a(), b()][], j=1..3), a(), c()][], i=1..3)] end: a:= n-> ring(3)[modp(n-1, 24)+1]: seq(a(n), n=1..99); # _Alois P. Heinz_, Aug 19 2008
%t ring[k_] := ring[k] = Module[{l, a, b, c, swap, h}, l = Range[4]; swap[i_, j_] := (h = l[[i]]; l[[i]] = l[[j]]; l[[j]] = h); a := (swap[1, 2]; swap[3, 4]; l[[k]]); b := (swap[2, 3]; l[[k]]); c := (swap[3, 4]; l[[k]] ); Join[{l[[k]]}, Flatten @ Table[ Join[ Flatten @ Table[{a, b}, {j, 1, 3}], {a}, {c}], {i, 1, 3}]]]; a[n_] := ring[3][[Mod[n-1, 24]+1]]; Table[a[n], {n, 1, 99}] (* _Jean-François Alcover_, Mar 18 2017, after _Alois P. Heinz_ *)
%Y Cf. A090277-A090284.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 24 2004
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