This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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:=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. = 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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