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%I #39 Oct 05 2022 07:45:37
%S 1,2,4,6,11,21,39,71,141,276,542,1070,2110,4189,8351,16618,33134,
%T 66129,131937,263483,526453,1051984,2102582,4203177,8403116,16800894,
%U 33593742,67174863,134328816,268624026,537192064,1074288649,2148414285,4296543181,8592585289
%N Number of gap-free compositions of n.
%C A gap-free composition contains all the parts between its smallest and largest part. a(5)=11 because we have: 5, 3+2, 2+3, 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
%H Alois P. Heinz, <a href="/A107428/b107428.txt">Table of n, a(n) for n = 1..1000</a>
%H Alois P. Heinz, <a href="/A107428/a107428.jpg">Plot of (a(n)-2^(n-2))/2^(n-2) for n = 42..1000</a>
%H P. Hitczenko and A. Knopfmacher, <a href="http://dx.doi.org/10.1016/j.disc.2005.02.008">Gap-free compositions and gap-free samples of geometric random variables</a>, Discrete Math., 294 (2005), 225-239.
%F a(n) ~ 2^(n-2). - _Alois P. Heinz_, Dec 07 2014
%e From _Gus Wiseman_, Oct 04 2022: (Start)
%e The a(0) = 1 through a(5) = 11 gap-free compositions:
%e () (1) (2) (3) (4) (5)
%e (11) (12) (22) (23)
%e (21) (112) (32)
%e (111) (121) (122)
%e (211) (212)
%e (1111) (221)
%e (1112)
%e (1121)
%e (1211)
%e (2111)
%e (11111)
%e (End)
%p b:= proc(n, i, t) option remember; `if`(n=0, t!,
%p `if`(i<1 or n<i, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))
%p end:
%p a:= n-> add(b(n, i, 0), i=1..n):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Apr 14 2014
%t Table[Length[Select[Level[Map[Permutations,IntegerPartitions[n]],{2}],Length[Union[#]]==Max[#]-Min[#]+1&]],{n,1,20}] (* _Geoffrey Critzer_, Apr 13 2014 *)
%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, 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_ *)
%Y The unordered version (partitions) is A034296, ranked by A073491.
%Y The initial case is A107429, unordered A000009, ranked by A333217.
%Y The unordered complement is counted by A239955, ranked by A073492.
%Y These compositions are ranked by A356841.
%Y The complement is counted by A356846, ranked by A356842
%Y A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356233 counts factorizations into gapless numbers.
%Y Cf. A055932, A073493, A137921, A251729, A356224, A356843, A356845.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, May 26 2005
%E More terms from _Vladeta Jovovic_, May 26 2005
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