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.
László Németh, James A. Sellers, and László Szalay, Extending Arithmetic Properties for Compositions Wherein Parts That Are Non-Multiples of k Can Be of t Kinds, Period. Math. Hung. (2026). See p. 2.
Stefan Schuster, Maximilian Fichtner, and Severin Sasso, Use of Fibonacci numbers in lipidomics - Enumerating various classes of fatty acids, Sci. Rep. 7 (2017), Art. 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);
# Alternative:
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
(SageMath)
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
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
More terms from James Sellers, Jun 05 2000
Better description from Emeric Deutsch, Apr 09 2005
STATUS
approved