A052945 Number of compositions of n when each odd part can be of two kinds.

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

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

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

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1004
• S. Schuster, M. Fichtner and S. Sasso, Use of Fibonacci numbers in lipidomics - Enumerating various classes of fatty acids, Sci. Rep., 7 (2017) 39821.
• Index entries for linear recurrences with constant coefficients, signature (2,2).

Crossrefs

Row sums of A105474.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-2*x-2*x^2) )); // G. C. Greubel, Oct 18 2019

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 *)

PARI

Vec((x-1)*(1+x)/(-1+2*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

Sage

def A052945_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2)/(1-2*x-2*x^2) ).list()
A052945_list(30) # G. C. Greubel, Oct 18 2019

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)

Extensions

More terms from James Sellers, Jun 05 2000

Better description from Emeric Deutsch, Apr 09 2005