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A251729 Number of gap-free but not complete compositions of n. 5
0, 1, 1, 2, 3, 3, 6, 6, 14, 12, 27, 33, 58, 86, 134, 210, 323, 539, 810, 1371, 2044, 3510, 5263, 8927, 13702, 22870, 35821, 58750, 93343, 152236, 243244, 395078, 634342, 1027876, 1656543, 2676693, 4325727, 6982440, 11299457, 18232217, 29518334, 47641410 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

A composition is gap-free but not complete if all integers in the interval defined by the smallest and the largest part are parts but 1 is not a part.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Alois P. Heinz, Plot of (a(n)+a(n+1)-a(n+2))/a(n+2) for n = 150..1000

P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.

FORMULA

a(n) = A107428(n) - A107429(n).

lim_{n -> oo} a(n)/a(n-1) = (1+sqrt(5))/2 = phi = A001622.

EXAMPLE

a(6) = 3: [6], [3,3], [2,2,2].

a(7) = 6: [7], [3,4], [4,3], [2,2,3], [2,3,2], [3,2,2].

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, 0, t!),

     `if`(i<1 or n<i, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))

    end:

a:= n-> add(b(n, i, 0), i=1..n):

seq(a(n), n=1..50);

CROSSREFS

Cf. A001622, A034296, A107428, A107429, A188575, A238353, A264396.

Sequence in context: A301703 A143715 A159685 * A187763 A187262 A117670

Adjacent sequences:  A251726 A251727 A251728 * A251730 A251731 A251732

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 07 2014

STATUS

approved

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Last modified September 28 13:24 EDT 2020. Contains 337393 sequences. (Running on oeis4.)