A060112 Sums of nonconsecutive factorial numbers.

0, 1, 2, 6, 7, 24, 25, 26, 120, 121, 122, 126, 127, 720, 721, 722, 726, 727, 744, 745, 746, 5040, 5041, 5042, 5046, 5047, 5064, 5065, 5066, 5160, 5161, 5162, 5166, 5167, 40320, 40321, 40322, 40326, 40327, 40344, 40345, 40346, 40440, 40441, 40442

Offset: 1

Author

Antti Karttunen, Mar 01 2001

Keywords

• nonn
• easy
• nice

Comments

Zeckendorf (Fibonacci) expansion of n (A003714) reinterpreted as a factorial expansion.

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.

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.

Examples

			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.,
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Arthur T. White, Ringing the Changes, Math. Proc. Camb. Phil. Soc., September 1983, Vol. 94, part 2, pp. 203-215.
• Index entries for sequences related to bell ringing

Crossrefs

Subset of A059590. Cf. also A001611, A064640.

For PermRevLexRank, see A056019, for fibbinary see A048679 and A003714.

Programs

Maple

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;

Mathematica

With[{b = MixedRadix[Range[12, 2, -1]]}, FromDigits[#, b] & /@ Select[Tuples[{0, 1}, 8], SequenceCount[#, {1, 1}] == 0 &]] (* Michael De Vlieger, Jun 26 2017 *)

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)
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

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

Formula

a(n) = PermRevLexRank(CampanoPerm(n))

a(A001611(n)) = (n-1)! for n > 2. - David A. Corneth, Jun 25 2017