login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099087 Expansion of 1/(1 - 2*x + 2*x^2). 16
1, 2, 2, 0, -4, -8, -8, 0, 16, 32, 32, 0, -64, -128, -128, 0, 256, 512, 512, 0, -1024, -2048, -2048, 0, 4096, 8192, 8192, 0, -16384, -32768, -32768, 0, 65536, 131072, 131072, 0, -262144, -524288, -524288, 0, 1048576, 2097152, 2097152, 0, -4194304, -8388608, -8388608, 0, 16777216 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Yet another variation on A009545.

Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(Pi*n/2). See A009116.

Binomial transform of A057077. - R. J. Mathar, Nov 04 2008

Partial sums of A146559. - Philippe Deléham, Dec 01 2008

Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

Also the inverse Catalan transform of A000079. - Arkadiusz Wesolowski, Oct 26 2012

LINKS

Table of n, a(n) for n=0..48.

Index entries for linear recurrences with constant coefficients, signature (2,-2)

FORMULA

E.g.f.: exp(x)(cos(x)+sin(x)); a(n)=2^(n/2)(cos(Pi*n/4)+sin(Pi*n/4)); a(n)=Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, k-i)binomial(n, i)(-1)^(k-i); a(n)=2a(n-1)-2a(n-2).

a(n) = (1-i)^(n-1)+(1+i)^(n-1) where i=sqrt(-1). a(n) = 2 Sum_{k=0..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. - R. J. Mathar, Apr 18 2008

a(n)=Sum_{k=0..n} A109466(n,k)*2^k. - Philippe Deléham, Oct 28 2008

E.g.f.: (cos(x)+sin(x))*exp(x)=G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011

G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012

a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - Philippe Deléham, Feb 13 2013

MATHEMATICA

CoefficientList[Series[1/(1 - 2 x + 2 x^2), {x, 0, 48}], x] (* Michael De Vlieger, Dec 24 2015 *)

PROG

(Sage) [lucas_number1(n, 2, 2) for n in xrange(1, 50)] # Zerinvary Lajos, Apr 23 2009

(PARI) x='x+O('x^100); Vec(1/(1-2*x+2*x^2)) \\ Altug Alkan, Dec 24 2015

CROSSREFS

Cf. A009545, A146559.

Sequence in context: A194656 A283240 A108520 * A009545 A084102 A221609

Adjacent sequences:  A099084 A099085 A099086 * A099088 A099089 A099090

KEYWORD

easy,sign

AUTHOR

Paul Barry, Sep 24 2004

EXTENSIONS

Signs added by N. J. A. Sloane, Nov 14 2006

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 23 00:16 EDT 2017. Contains 290953 sequences.