|
| |
|
|
A009545
|
|
E.g.f. sin(x)*exp(x).
|
|
35
|
|
|
|
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 plan 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 DELEHAM, 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
|
|
|
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:# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2009]
restart: 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 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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}]] (*From 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] (* From Harvey P. Dale, Oct 13 2011 *)
|
|
|
PROG
|
(Sage) [lucas_number1(n, 2, 2) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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 */
|
|
|
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.
A000225, A144081 [From Gary W. Adamson, Sep 10 2008]
A146559
Sequence in context: A108520 A099087 * A084102 A221609 A160125 A151868
Adjacent sequences: A009542 A009543 A009544 * A009546 A009547 A009548
|
|
|
KEYWORD
|
sign,easy,nice
|
|
|
AUTHOR
|
R. H. Hardin
|
|
|
EXTENSIONS
|
Extended with signs Mar 15 1997 by Olivier Gerard.
More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000.
Definition corrected by Joerg Arndt, Apr 24 2011.
|
|
|
STATUS
|
approved
|
| |
|
|
