0,3

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 (qntmpkt(AT)yahoo.com), Sep 10 2008]

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.

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

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

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 (mlb(AT)well.com)

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)/(2i) =2a(n-1)-2a(n-2) [with a(0)=0 and a(1)=1] - Henry Bottomley (se16(AT)btinternet.com), May 10 2001

a(n-1)=(1+I)^n+(1-I)^n - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 28 2002

a(n)=sum(k=0, n-1, (-1)^floor(k/2)*C(n-1, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 31 2003

a(n)=2^(n/2)sin(pi*n/4) - Paul Barry (pbarry(AT)wit.ie), Sep 17 2003

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

a(n+1)=Sum_{k, 0<=k<=n}2^k*A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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)=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

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]

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

(Other) sage: [lucas_number1(n, 2, 2) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]

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 (qntmpkt(AT)yahoo.com), Sep 10 2008]

Sequence in context: A072690 A108520 A099087 this_sequence A084102 A160125 A151868

Adjacent sequences: A009542 A009543 A009544 this_sequence A009546 A009547 A009548

sign,easy,nice

R. H. Hardin (rhhardin(AT)att.net)

Extended with signs Mar 15 1997 by Olivier Gerard. More terms from Larry Reeves (larryr(AT)acm.org), Aug 24 2000