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A060112 Sums of nonconsecutive factorial numbers.

%I A060112 #19 Jun 27 2017 06:23:55
%S A060112 0,1,2,6,7,24,25,26,120,121,122,126,127,720,721,722,726,727,744,745,
%T A060112 746,5040,5041,5042,5046,5047,5064,5065,5066,5160,5161,5162,5166,5167,
%U A060112 40320,40321,40322,40326,40327,40344,40345,40346,40440,40441,40442
%N A060112 Sums of nonconsecutive factorial numbers.
%C A060112 Zeckendorf (Fibonacci) expansion of n (A003714) reinterpreted as a factorial expansion.
%C A060112 Also positions in A055089, A060117 and A060118 of the permutations that are composed of disjoint adjacent transpositions only. (That these positions are same can be seen by comparing algorithms PermRevLexUnrankAMSD, PermUnrank3R, PermUnrank3L in the respective sequences). Thus also positions of the fixed terms in A065181-A065184. See comment at A065163.
%C A060112 Written as disjoint cycles the permutations are: (), (1 2), (2 3), (3 4), (1 2)(3 4), (4 5), (1 2)(4 5), (2 3)(4 5), etc. Apart from the first one (the identity), these are the only kind of permutations used in campanology when moving from one "change" to next.
%H A060112 Charles R Greathouse IV, <a href="/A060112/b060112.txt">Table of n, a(n) for n = 1..10000</a>
%H A060112 Arthur T. White, <a href="https://doi.org/10.1017/S0305004100061053">Ringing the Changes</a>, Math. Proc. Camb. Phil. Soc., September 1983, Vol. 94, part 2, pp. 203-215.
%H A060112 <a href="/index/Be#bell_ringing">Index entries for sequences related to bell ringing</a>
%F A060112 a(n) = PermRevLexRank(CampanoPerm(n))
%F A060112 a(A001611(n)) = (n-1)! for n > 2. - _David A. Corneth_, Jun 25 2017
%e A060112 Zeckendorf Expansions of first few natural numbers and the corresponding values when interpreted as factorial expansions: 0 = 0 = 0, 1 = 1 = 1, 2 = 10 = 2, 3 = 100 = 6, 4 = 101 = 7, 5 = 1000 = 24, 6 = 1001 = 25, 7 = 1010 = 26, 8 = 10000 = 120, etc.,
%p A060112 CampanoPerm := proc(n) local z,p,i; p := []; z := fibbinary(n); i := 1; while(z > 0) do if(1 = (z mod 2)) then p := permul(p,[[i,i+1]]); fi; i := i+1; z := floor(z/2); od; RETURN(convert(p,'permlist',i)); end;
%t A060112 With[{b = MixedRadix[Range[12, 2, -1]]}, FromDigits[#, b] & /@ Select[Tuples[{0, 1}, 8], SequenceCount[#, {1, 1}] == 0 &]] (* _Michael De Vlieger_, Jun 26 2017 *)
%o A060112 (PARI) fill(lim,k,val)=if(k>#f, return); my(t=val+f[k]); if(t<=lim, listput(v,t); fill(lim,k+2,t)); fill(lim,k+1,val)
%o A060112 list(lim)=my(k,t=1); local(f=List(),v=List([0])); while((t*=k++)<=lim, listput(f,t)); f=Vecrev(f); fill(lim,1,0); Set(v) \\ _Charles R Greathouse IV_, Jun 25 2017
%o A060112 (PARI) first(n) = my(res = [0, 1], k = 1, t = 1, p = 1); while(#res < n, k++; t++; p *= t; res = concat(res, vector(fibonacci(k), i, res[i]+p))); vector(n, i, res[i]) \\ _David A. Corneth_, Jun 26 2017
%Y A060112 Subset of A059590. Cf. also A001611, A064640.
%Y A060112 For PermRevLexRank, see A056019, for fibbinary see A048679 and A003714.
%K A060112 nonn,easy,nice
%O A060112 1,3
%A A060112 _Antti Karttunen_, Mar 01 2001