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A188575 Number of non-complete compositions of n.

%I A188575 #27 May 18 2020 14:56:17
%S A188575 0,1,1,4,8,14,31,63,129,248,509,1011,2044,4089,8167,16360,32725,65482,
%T A188575 131017,262176,524167,1048678,2096985,4194358,8387802,16776408,
%U A188575 33550943,67101615,134199983,268399122,536793004,1073590077,2147187353,4294419287,8588940438
%N A188575 Number of non-complete compositions of n.
%H A188575 Alois P. Heinz, <a href="/A188575/b188575.txt">Table of n, a(n) for n = 1..1000</a>
%H A188575 M. Archibald and A. Knopfmacher, <a href="https://doi.org/10.1016/j.disc.2011.01.012">The largest missing value in a composition of an integer</a>, Discrete Math., 311 (2011), 723-731.
%H A188575 Alois P. Heinz, <a href="/A188575/a188575.jpg">Plot of (a(n)-2^(n-2))/2^(n-2) for n = 60..1000</a>
%F A188575 a(n) = 2^(n-1) - A107429(n) ~ 2^(n-2). - _Alois P. Heinz_, Dec 06 2014
%p A188575 b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
%p A188575       `if`(i<1 or n<i, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))
%p A188575     end:
%p A188575 a:= n-> 2^(n-1) -add(b(n, i, 0), i=1..n):
%p A188575 seq(a(n), n=1..40);  # _Alois P. Heinz_, Dec 06 2014
%t A188575 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_] := 2^(n-1)-Sum[b[n, i, 0], {i, 1, n} ]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%Y A188575 Cf. A107429, A188576, A188577, A251729.
%K A188575 nonn
%O A188575 1,4
%A A188575 _N. J. A. Sloane_, Apr 04 2011
%E A188575 More terms from _Alois P. Heinz_, Dec 06 2014