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

0,2

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

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).

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

Adjacent sequences: A052942 A052943 A052944 * A052946 A052947 A052948

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