%I A009545
%S A009545 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,
%T A009545 4096,8192,8192,0,16384,32768,32768,0,65536,131072,131072,0,262144,524288,
%U A009545 524288,0,1048576,2097152,2097152,0,4194304,8388608,8388608,0,16777216,
               33554432
%V A009545 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,
%W A009545 4096,8192,8192,0,-16384,-32768,-32768,0,65536,131072,131072,0,-262144,
               -524288,
%X A009545 -524288,0,1048576,2097152,2097152,0,-4194304,-8388608,-8388608,0,16777216,
               33554432
%N A009545 Expansion of sin(x)*exp(x).
%C A009545 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
%C A009545 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 (qntmpkt(AT)yahoo.com), Sep 10 2008]
%D A009545 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A009545 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal 
               Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A009545 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.
%D A009545 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.
%H A009545 T. D. Noe, <a href="b009545.txt">Table of n, a(n) for n=0..100</a>
%H A009545 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A009545 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A009545 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
%F A009545 Imaginary part of (1+i)^n - Marc LeBrun (mlb(AT)well.com)
%F A009545 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.
%F A009545 a(n) =((1+i)^n-(1-i)^n)/(2i) =2a(n-1)-2a(n-2) [with a(0)=0 and a(1)=1] 
               - Henry Bottomley (se16(AT)btinternet.com), May 10 2001
%F A009545 a(n-1)=(1+I)^n+(1-I)^n - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 
               28 2002
%F A009545 a(n)=sum(k=0, n-1, (-1)^floor(k/2)*C(n-1, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jan 31 2003
%F A009545 a(n)=2^(n/2)sin(pi*n/4) - Paul Barry (pbarry(AT)wit.ie), Sep 17 2003
%F A009545 a(n)=sum{k=0..floor(n/2), C(n, 2k+1)(-1)^k} - Paul Barry (pbarry(AT)wit.ie), 
               Sep 20 2003
%F A009545 a(n+1)=Sum_{k, 0<=k<=n}2^k*A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 13 2006
%F A009545 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
%F A009545 a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3, which implies the sequence is identical 
               to its fourth differences. Binomial transform of 0, 1, 0, -1. - Paul 
               Curtz (bpcrtz(AT)free.fr), Dec 21 2007
%p A009545 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]
%p A009545 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]
%t A009545 nn=104; Range[0,nn-1]! CoefficientList[Series[Sin[x]Exp[x], {x,0,nn}], 
               x] - from T. D. Noe, May 26 2007
%o A009545 (Other) sage: [lucas_number1(n,2,2) for n in xrange(0, 51)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A009545 Cf. A009116. For minor variants of this sequence see A108520, A084102, 
               A099087.
%Y A009545 a(2*n)= A056594(n)*2^n, n >= 1, a(2*n+1)= A057077(n)*2^n.
%Y A009545 This is the next term in the sequence A015518, A002605, A000129, A000079, 
               A001477.
%Y A009545 A000225, A144081 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 10 
               2008]
%Y A009545 Sequence in context: A072690 A108520 A099087 this_sequence A084102 A160125 
               A151868
%Y A009545 Adjacent sequences: A009542 A009543 A009544 this_sequence A009546 A009547 
               A009548
%K A009545 sign,easy,nice
%O A009545 0,3
%A A009545 R. H. Hardin (rhhardin(AT)att.net)
%E A009545 Extended with signs Mar 15 1997 by Olivier Gerard. More terms from Larry 
               Reeves (larryr(AT)acm.org), Aug 24 2000