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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 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.)