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 A107429 Number of complete compositions of n. 43
 1, 1, 3, 4, 8, 18, 33, 65, 127, 264, 515, 1037, 2052, 4103, 8217, 16408, 32811, 65590, 131127, 262112, 524409, 1048474, 2097319, 4194250, 8389414, 16778024, 33557921, 67116113, 134235473, 268471790, 536948820, 1073893571, 2147779943, 4295515305, 8590928746 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A composition is complete if it is gap-free and contains a 1. - Geoffrey Critzer, Apr 13 2014 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 70 terms from Daniel Reimhult) Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 40..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) ~ 2^(n-2). - Vaclav Kotesovec, Sep 05 2014 EXAMPLE a(5)=8 because we have: 2+2+1, 2+1+2, 1+2+2, 2+1+1+1, 1+2+1+1, 1+1+2+1, 1+1+1+2, 1+1+1+1+1. - Geoffrey Critzer, Apr 13 2014 MAPLE b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0), `if`(i<1 or n add(b(n, i, 0), i=1..n): seq(a(n), n=1..40); # Alois P. Heinz, Apr 14 2014 MATHEMATICA Table[Length[Select[Level[Map[Permutations, IntegerPartitions[n]], {2}], MemberQ[#, 1]&&Length[Union[#]]==Max[#]-Min[#]+1&]], {n, 1, 20}] (* Geoffrey Critzer, Apr 13 2014 *) b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, t!, 0], If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *) CROSSREFS Cf. A107428, A034296, A188575, A251729. Sequence in context: A215095 A192474 A183494 * A061273 A254715 A107328 Adjacent sequences: A107426 A107427 A107428 * A107430 A107431 A107432 KEYWORD nonn AUTHOR N. J. A. Sloane, May 26 2005 EXTENSIONS More terms from Vladeta Jovovic, May 26 2005 STATUS approved

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Last modified March 4 23:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)