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A052945 Number of compositions of n when each odd part can be of two kinds. 9
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 (list; graph; refs; listen; history; text; internal format)
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 A047086 A006574

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

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)