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A052945
Number of compositions of n when each odd part can be of two kinds.
10
1, 2, 5, 14, 38, 104, 284, 776, 2120, 5792, 15824, 43232, 118112, 322688, 881600, 2408576, 6580352, 17977856, 49116416, 134188544, 366609920, 1001596928, 2736413696, 7476021248, 20424869888, 55801782272, 152453304320
OFFSET
0,2
COMMENTS
Also number of compositions of n into 2 sorts of parts where the kinds of parts are unordered inside a run of identical parts, see example. Replacing "unordered" by "ordered" gives A025192. - Joerg Arndt, Apr 28 2013
Numbers of straight-chain fatty acids involving oxo groups (or hydroxy groups), if cis-/trans isomerism is considered while stereoisomerism is neglected. - Stefan Schuster, Apr 19 2018
FORMULA
G.f.: (1 - x)*(1 + x)/(1 - 2*x - 2*x^2).
a(n) = 2*(a(n-1) + a(n-2)).
a(n) = Sum_{alpha=RootOf(-1+2*z+2z^2)} alpha^(-1-n)/4.
From Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009: (Start)
a(n) = ((2+sqrt(3))*(1+sqrt(3))^(n-1) + (2-sqrt(3))*(1-sqrt(3))^(n-1))/2 for n>0.
First binomial transform of 2, 3, 6, 9, 18, 27, 54, 81, ... starting after 1. (End)
EXAMPLE
a(3)=14 because we have (3),(3'),(1,2),(1',2),(2,1),(2,1'),(1,1,1),(1,1,1'),(1,1',1),(1,1',1'),(1',1,1),(1',1,1'),(1',1',1) and (1',1',1').
There are a(3)=14 such compositions of 3. Here p:s stands for "part p of sort s":
01: [ 1:0 1:0 1:0 ]
02: [ 1:0 1:0 1:1 ]
03: [ 1:0 1:1 1:1 ]
04: [ 1:0 2:0 ]
05: [ 1:0 2:1 ]
06: [ 1:1 1:1 1:1 ]
07: [ 1:1 2:0 ]
08: [ 1:1 2:1 ]
09: [ 2:0 1:0 ]
10: [ 2:0 1:1 ]
11: [ 2:1 1:0 ]
12: [ 2:1 1:1 ]
13: [ 3:0 ]
14: [ 3:1 ]
- Joerg Arndt, Apr 28 2013
MAPLE
spec:= [S, {S=Sequence(Prod(Union(Sequence(Prod(Z, Z)), Sequence(Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-x^2)/(1-2*x-2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 18 2019
MATHEMATICA
LinearRecurrence[{2, 2, }, {1, 2, 5}, 30] (* G. C. Greubel, Oct 18 2019 *)
PROG
(PARI) Vec((x-1)*(1+x)/(-1+2*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-2*x-2*x^2) )); // G. C. Greubel, Oct 18 2019
(Sage)
def A052945_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x^2)/(1-2*x-2*x^2) ).list()
A052945_list(30) # G. C. Greubel, Oct 18 2019
(GAP) a:=[2, 5];; for n in [3..30] do a[n]:=2*(a[n-1]+a[n-2]); od; Concatenation([1], a); # G. C. Greubel, Oct 18 2019
CROSSREFS
Row sums of A105474.
Sequence in context: A292327 A084085 A052985 * A026288 A006574 A053419
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 05 2000
Better description from Emeric Deutsch, Apr 09 2005
STATUS
approved