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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)
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<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..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
Sequence in context: A215095 A192474 A183494 * A061273 A254715 A107328
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.)