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A009545 E.g.f. sin(x)*exp(x). 37
0, 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, 33554432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also first of the two associated sequences a(n) and b(n) built from a(0)=0 and a(1)=1 with the formulas a(n)=a(n-1)+b(n-1) b(n)=-a(n-1)+b(n-1). The initial terms of the second sequence b(n) are 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32, -64, -64, 0, 128, 256, ... The points Mn(a(n)+b(n)*I) of the complex plane are located on the spiral logarithmic rho=2*(1/2)^(2*theta)/Pi) and on the straight lines drawn from the origin with slopes: Infinity, 1/2, 0, -1/2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007

A000225: (1, 3, 7, 15, 31,...) = 2^n - 1 = INVERT transform of A009545 starting (1, 2, 2, 0, -4, -8,...). (Cf. comments in A144081). [From Gary W. Adamson, Sep 10 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

The variant 0, 1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64,. (with different signs) is the Lucas U(-2,2) sequence. - R. J. Mathar, Jan 08 2013

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

REFERENCES

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=2, q=-2.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38) and (45), lhs, m=2.

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

Wikipedia, Lucas sequence

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for Lucas sequences

FORMULA

a(0)=0; a(1)=1; a(2)=2; a(3)=2; a(n)=-4*a(n-4), n>3 - Larry Reeves (larryr(AT)acm.org), Aug 24 2000

Imaginary part of (1+i)^n - Marc LeBrun

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

E.g.f.: sin(x)*exp(x).

a(n)= S(n-1, sqrt(2))*(sqrt(2))^(n-1) with S(n, x)= U(n, x/2) Chebyshev's polynomials of the 2nd kind, Cf. A049310, S(-1, x) := 0.

a(n) =((1+i)^n-(1-i)^n)/(2*i) =2*a(n-1)-2*a(n-2) [with a(0)=0 and a(1)=1]. - Henry Bottomley, May 10 2001

a(n) = (1+i)^(n-2)+(1-i)^(n-2). - Benoit Cloitre, Oct 28 2002

a(n)=sum(k=0, n-1, (-1)^floor(k/2)*C(n-1, k)). - Benoit Cloitre, Jan 31 2003

a(n)=2^(n/2)sin(pi*n/4) - Paul Barry, Sep 17 2003

a(n)=sum{k=0..floor(n/2), C(n, 2*k+1)*(-1)^k}. - Paul Barry, Sep 20 2003

a(n+1)=Sum_{k, 0<=k<=n}2^k*A109466(n,k). - Philippe Deléham, Nov 13 2006

a(n) = 2*((1/2)^(2*theta(n)/pi))*cos(theta(n) where: theta(4*p+1)=p*Pi + Pi/2 theta(4*p+2)=p*Pi + Pi/4 theta(4*p+3)=p*Pi - Pi/4 theta(4*p+4)=p*Pi - Pi/2 or a(0)=0 a(1)=1 a(2)=2 a(3)=2 and for n>3 a(n)=-4*a(n-4). Same formulas for the second sequence replacing cosines by sines. For example: a(0) = 0 b(0) = 1 a(1) = 0+1 = 1 b(1) = -0+1 = 1 a(2) = 1+1 = 2 b(2) = -1+1 = 0 a(3) = 2+0 = 2 b(3) = -2+0 = -2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 30 2007

a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3), n > 3, which implies the sequence is identical to its fourth differences. Binomial transform of 0, 1, 0, -1. - Paul Curtz, Dec 21 2007

Logarithm g.f. atan(x/(1-x))=sum(n>0, a(n)/n*x^n). [From Vladimir Kruchinin, Aug 11 2010]

a(n) = A046978(n) * A016116(n). - Paul Curtz, Apr 24 2011

E.g.f.: exp(x) * sin(x) =x+x^2/(G(0)-x); G(k)=2k+1+x-x*(2k+1)/(4k+3+x+x^2*(4k+3)/( (2k+2)*(4k+5)-x^2-x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011

a(n) = Im( (1+i)^n ) where i=sqrt(-1). - Stanislav Sykora, Jun 11 2012.

G.f.: x*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

G.f.: G(0)*x/(2*(1-x)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013

G.f.: x + x^2*W(0), where W(k) = 1 + 1/(1 - x*(k+1)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 2*x)/( x*(4*k+4 - 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 06 2013

MAPLE

t1 := sum(n*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 50 do printf(`%d, `, coeff(F, x, i)) od:# Zerinvary Lajos, Mar 22 2009

G(x):=exp(x)*sin(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..50 ); # Zerinvary Lajos, Apr 05 2009

MATHEMATICA

nn=104; Range[0, nn-1]! CoefficientList[Series[Sin[x]Exp[x], {x, 0, nn}], x] - from T. D. Noe, May 26 2007

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

f[n_] := (1 + I)^(n - 2) + (1 - I)^(n - 2); Array[f, 51, 0] (* Robert G. Wilson v, May 30 2011 *)

LinearRecurrence[{2, -2}, {0, 1}, 110] (* Harvey P. Dale, Oct 13 2011 *)

PROG

(Sage) [lucas_number1(n, 2, 2) for n in xrange(0, 51)] # [Zerinvary Lajos, Apr 23 2009]

(Pari) x='x+O('x^66); /* that many terms */

Vec(serlaplace(exp(x)*sin(x))) /* show terms */ /* Joerg Arndt, Apr 24 2011 */

(Sage)

def A146559():

    x, y = 0, -1

    while true:

        yield x

        x, y = x - y, x + y

a = A146559(); [a.next() for i in range(40)]  # Peter Luschny, Jul 11 2013

CROSSREFS

Cf. A009116. For minor variants of this sequence see A108520, A084102, A099087.

a(2*n)= A056594(n)*2^n, n >= 1, a(2*n+1)= A057077(n)*2^n.

This is the next term in the sequence A015518, A002605, A000129, A000079, A001477.

Cf. A000225, A144081. - Gary W. Adamson, Sep 10 2008

Cf. A146559.

Sequence in context: A108520 A099087 * A084102 A221609 A160125 A151868

Adjacent sequences:  A009542 A009543 A009544 * A009546 A009547 A009548

KEYWORD

sign,easy,nice,changed

AUTHOR

R. H. Hardin

EXTENSIONS

Extended with signs Mar 15 1997 by Olivier Gérard.

More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000.

Definition corrected by Joerg Arndt, Apr 24 2011.

STATUS

approved

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Last modified December 20 12:55 EST 2014. Contains 252248 sequences.