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A060112 Sums of nonconsecutive factorial numbers. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

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

FORMULA

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

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

EXAMPLE

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.,

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 *)

PROG

(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

CROSSREFS

Subset of A059590. Cf. also A001611, A064640.

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

Sequence in context: A004791 A220946 A243795 * A057914 A216037 A250547

Adjacent sequences:  A060109 A060110 A060111 * A060113 A060114 A060115

KEYWORD

nonn,easy,nice

AUTHOR

Antti Karttunen, Mar 01 2001

STATUS

approved

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Last modified July 21 12:41 EDT 2017. Contains 289642 sequences.