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

 

Logo

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!)
A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2). 13
1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general, the expansion of (1-x)/(1-2x+(m+1)x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...]=: powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..3322

Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

Index entries for linear recurrences with constant coefficients, signature (2,-4).

FORMULA

a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). - Philippe Deléham, Nov 14 2008

From Philippe Deléham, Nov 14 2008: (Start)

a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)

a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008

a(n) = (1/2)*((1-i*sqrt(3))^n + (1+i*sqrt(3))^n), with n >= 0 and i=sqrt(-1). - Paolo P. Lava, Nov 18 2008

a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008

a(n) = -8 * a(n-3). - Paul Curtz, Apr 22 2011

From Sergei N. Gladkovskii, Jul 27 2012: (Start)

G.f.: G(0) where G(k)= 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction, 3rd kind, 4-step).

E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k)= 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction, 3rd kind, 4-step). (End)

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013

a(n) = A088138(n+1)-A088138(n). - R. J. Mathar, Mar 04 2018

a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019

MATHEMATICA

CoefficientList[Series[(1-x)/(1-2x+4x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, -4}, {1, 1}, 30] (* Harvey P. Dale, Nov 11 2014 *)

CROSSREFS

Cf. A088138, A104537, A128018.

Sequence in context: A079458 A281914 A290378 * A128018 A104537 A019240

Adjacent sequences:  A138227 A138228 A138229 * A138231 A138232 A138233

KEYWORD

easy,sign

AUTHOR

Paul Barry, Mar 06 2008

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)