OFFSET
0,5
COMMENTS
Previous name was: Counts members of A056808 by number of factors.
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014
LINKS
FORMULA
a(n) = 2*a(n-1) + A117989(n-1). - Bob Selcoe, Apr 11 2014
G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020
EXAMPLE
A011782 begins 1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins 1 1 2 3 5 7 11 15 22 30 ...;
so sequence begins 0 0 0 1 3 9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - Bob Selcoe, Jul 08 2014
MAPLE
a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
seq(a(n), n=0..37); # Alois P. Heinz, Jan 30 2020
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !GreaterEqual@@#&]], {n, 0, 10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
a /@ Range[0, 37] (* Jean-François Alcover, May 23 2021 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Aug 29 2000
EXTENSIONS
More terms from James Sellers, Aug 31 2000
New name from Joerg Arndt, Sep 02 2013
STATUS
approved
