|
| |
|
|
A146559
|
|
Expansion of (1-x)/(1-2*x+2*x^2) .
|
|
21
|
|
|
|
1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, 256, 0, -512, -1024, -1024, 0, 2048, 4096, 4096, 0, -8192, -16384, -16384, 0, 32768, 65536, 65536, 0, -131072, -262144, -262144, 0, 524288, 1048576, 1048576, 0, -2097152, -4194304
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Partial sums of this sequence give A099087 . [From Philippe DELEHAM, Dec 01 2008]
Triangle
( 1)..
..( 1)..
2....( 0)....
..2....(-2)...
4....0....(-4)...
..4....-4...(-4)...
8....0....-8...( 0)....
..8....-8...-8...( 8)....
16...0....-16..0....(16)...
..16...-16..-16..16...(16)...
32...0....-32..0....32...(0)....
..32...-32..-32..32...32...(-32)..
64...0....-64..0....64...0....(-64)..
(1+i)^n = a(n) + A009545(n)*i where i = sqrt(-1).Philippe Deléham, Feb 13 2013
Row sums of triangle in A104597.-Philippe Deléham, Feb 20 2013
|
|
|
LINKS
|
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (2,-2).
|
|
|
FORMULA
|
a(0)=1, a(1)=1, a(n)=2*a(n-1)-2*a(n-2) for n>1.
a(n) = Sum_{k, 0<=k<=n} A124182(n,k)*(-2)^(n-k).
a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*(-1)^(n-k). [From Philippe DELEHAM, Nov 14 2008]
a(n)=(1/2)*[(1-I)^n+(1+I)^n], with n>=0 and I=sqrt(-1) [From Paolo P. Lava, Nov 18 2008]
a(n)=(-1)^n*A009116(n). [From Philippe DELEHAM, Dec 01 2008]
E.g.f.: exp(x)*cos(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
E.g.f.: cos(x)*exp(x) =1+x/(G(0)-x) where G(k)=4*k+1+x+(x^2)*(4*k+1)/((2*k+1)*(4*k+3)-(x^2)-x*(2*k+1)*(4*k+3)/( 2*k+2+x-x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
a(n) = Re( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012.
G.f.: 1 / (1 - x / (1 + x / (1 - 2*x))) = 1 + x / (1 + 2*x^2 / (1 - 2*x)). - Michael Somos, Jan 03 2013
|
|
|
EXAMPLE
|
1 + x - 2*x^3 - 4*x^4 - 4*x^5 + 8*x^7 + 16*x^8 + 16*x^9 - 32*x^11 - 64*x^12 - ...
|
|
|
MAPLE
|
G(x):=exp(x)*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..44 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]
|
|
|
MATHEMATICA
|
CoefficientList[Series[(1-x)/(1-2x+2x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{2, -2}, {1, 1}, 50] (* From Harvey P. Dale, Oct 13 2011 *)
|
|
|
PROG
|
(PARI) Vec((1-x)/(1-2x+2x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 11 2012
|
|
|
CROSSREFS
|
Cf. A009116, A009545.
Sequence in context: A195479 A112793 A009116 * A118434 A090132 A199051
Adjacent sequences: A146556 A146557 A146558 * A146560 A146561 A146562
|
|
|
KEYWORD
|
sign,easy
|
|
|
AUTHOR
|
Philippe DELEHAM, Nov 01 2008
|
|
|
STATUS
|
approved
|
| |
|
|
