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

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A088138 Generalized Gaussian Fibonacci integers. 14
0, 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, -1024, 0, 4096, 8192, 0, -32768, -65536, 0, 262144, 524288, 0, -2097152, -4194304, 0, 16777216, 33554432, 0, -134217728, -268435456, 0, 1073741824, 2147483648, 0, -8589934592, -17179869184, 0, 68719476736, 137438953472 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequence 0,1,-2,0,8,-16,... has g.f. x/(1+2*x-4*x^2), a(n)=2^n*sin(2n*pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)*x)/sqrt(3): 0,1,-3,0,9,...

a(n+1) is the Hankel transform of A100192. - Paul Barry, Jan 11 2007

a(n+1) is the Trinomial transform of A010892: a(n+1) = Sum[Trinomial[n,k]A010892[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907); - Paul Barry, Sep 10 2007

a(n+1) is the Hankel transform of A100067. [From Paul Barry, Jun 16 2009]

1) a(n)=A131577*A128834 2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244. 3) Sequence is identical to every 2n-th differences divided by (-3)^n. 4) a(3n)+a(3n+1)+a(3n+2)=3,-24,192,=3*A001018 signed.

5) For missing terms in a(n) see A013731=4*A001018. [From Paul Curtz, Oct 04 2009]

The coefficient of i of Q^n, where Q is the quaternion 1+i+j+k. Due to symmetry, also the coefficients of j and of k. - Stanislav Sykora, Jun 11 2012.[The coefficients of 1 are in A138230.- Wolfdieter Lang, Jan 28 2016]

With different signs, 0, 1, -2, 0, 8, -16, 0, 64, -128, 0, 512, -1024,... is the Lucas U(-2,4) sequence. - R. J. Mathar, Jan 08 2013

LINKS

Robert Israel, Table of n, a(n) for n = 0..3300

Wikipedia, Lucas sequence

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

Index entries for Lucas sequences

FORMULA

G.f.: x/(1-2*x+4*x^2)

E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3)

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

a(n) = ((1+i*sqrt(3))^n-(1-i*sqrt(3))^n)/(2*i*sqrt(3))

a(n) = Im( (1+i*sqrt(3))^n/sqrt(3) ).

a(n) = sum(k=0..floor(n/2), C(n, 2*k+1)*(-3)^k ).

a(n) = a(n-1)+a(n-2)+2*a(n-3); a(n) = 2*a(n-1)-a(n-2)+2*a(n-3); a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-4). - Paul Curtz, Oct 04 2009

E.g.f. exp(x)*sin(sqrt(3)*x)/sqrt(3) = G(0)*x^2 where G(k)= 1 + (3*k+2)/(2*x - 32*x^5/( 16*x^4 - 3*(k+1)*(3*k+2)*(3*k+4)*(3*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2012

G.f.: x/(1-2*x+4*x^2) = 2*x^2*G(0) where G(k)= 1 + 1/(2*x - 32*x^5/(16*x^4 - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jul 27 2012

MAPLE

M:= <<1+I, 1+I>|<I-1, 1-I>>:

T:= <<-I/2, 0>|<0, I/2>>:

seq(LinearAlgebra:-Trace(T.M^n), n=0..100); # Robert Israel, Jan 28 2016

MATHEMATICA

Join[{a=0, b=1}, Table[c=2*b-4*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)

LinearRecurrence[{2, -4}, {0, 1}, 40] (* Vincenzo Librandi, Jan 29 2016 *)

PROG

(Sage) [lucas_number1(n, 2, 4) for n in xrange(0, 39)] # Zerinvary Lajos, Apr 23 2009

(PARI) /* lists powers of any quaternion) */

QuaternionToN(a, b, c, d, nmax) = {local (C); C = matrix(nmax+1, 4); C[1, 1]=1; for(n=2, nmax+1, C[n, 1]=a*C[n-1, 1]-b*C[n-1, 2]-c*C[n-1, 3]-d*C[n-1, 4]; C[n, 2]=b*C[n-1, 1]+a*C[n-1, 2]+d*C[n-1, 3]-c*C[n-1, 4]; C[n, 3]=c*C[n-1, 1]-d*C[n-1, 2]+a*C[n-1, 3]+b*C[n-1, 4]; C[n, 4]=d*C[n-1, 1]+c*C[n-1, 2]-b*C[n-1, 3]+a*C[n-1, 4]; ); return (C); } /* Stanislav Sykora, Jun 11 2012 */

CROSSREFS

Cf. A084102, A088137, A045873, A088139, A138230.

Sequence in context: A009794 A171402 A104506 * A186033 A120559 A120555

Adjacent sequences:  A088135 A088136 A088137 * A088139 A088140 A088141

KEYWORD

easy,sign

AUTHOR

Paul Barry, Sep 20 2003

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 December 6 06:58 EST 2016. Contains 278775 sequences.