|
|
A052910
|
|
Expansion of 1 + 2/(1-2*x-x^3).
|
|
3
|
|
|
1, 2, 4, 8, 18, 40, 88, 194, 428, 944, 2082, 4592, 10128, 22338, 49268, 108664, 239666, 528600, 1165864, 2571394, 5671388, 12508640, 27588674, 60848736, 134206112, 296000898, 652850532, 1439907176, 3175815250, 7004481032
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1-x^3)/(1-2*x-x^3).
a(n) = 2*a(n-1) + a(n-3), with a(0)=1, a(1)=2, a(2)=4, a(3)=8.
a(n) = Sum_{alpha=RootOf(-1 + 2*z + z^3)} (2/59)*(12 -8*alpha + 9*alpha^2)*alpha^(-1-n).
|
|
MAPLE
|
spec := [S, {S=Sequence(Prod(Sequence(Prod(Z, Z, Z)), Union(Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
|
|
MATHEMATICA
|
Join[{1}, LinearRecurrence[{2, 0, 1}, {2, 4, 8}, 30]] (* Harvey P. Dale, Jun 07 2012 *)
|
|
PROG
|
(PARI) my(x='x+O('x^30)); Vec((1-x^3)/(1-2*x-x^3)) \\ G. C. Greubel, Oct 15 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^3)/(1-2*x-x^3) )); // G. C. Greubel, Oct 15 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-x^3)/(1-2*x-x^3)).list()
(GAP) a:=[2, 4, 8];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Oct 15 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|