0,1

Any sequence b(n) satisfying the recurrence b(n)=b(n-1)-b(n-2) can be written as b(n)=b(0)*a(n)+(b(1)-b(0))*a(n-1).

a(n) is the determinant of the n X n matrix M with m(i,j)=1 if |i-j| <= 1 and 0 otherwise. - Mario Catalani (mario.catalani(AT)unito.it), Jan 25 2003

Also row sum of triangle in A108299; a(n)=L(n-1,1), where L is also defined as in A108299; see A061347 for L(n,-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005

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

Index entries for sequences related to linear recurrences with constant coefficients

Ralph E. Griswold, Shaft Sequences

Index entries for sequences related to Chebyshev polynomials.

G.f.: 1/(1-x+x^2); a(n)=a(n-1)-a(n-2), a(0)=1, a(1)=1; a(n)=((-1)^Floor(n/3) + (-1)^Floor((n+1)/3))/2.

a(n)= 0 if n mod 6 = 2 or 5, a(n)= + 1 if n mod 6 = 0 or 1, a(n)= -1 else. a(n)= S(n, 1) = U(n, 1/2) (Chebyshev U(n, x) polynomials).

a(n) = sqrt(4/3)*Imaginary[(1/2+i*sqrt(3/4))^(n+1)] - Henry Bottomley (se16(AT)btinternet.com), Apr 12 2000

Binomial transform of A057078. a(n)=sum{k=0..n, C(k, n-k)(-1)^(n-k) }. - Paul Barry (pbarry(AT)wit.ie), Sep 13 2003

a(n)=2sin(pi*n/3+pi/3)/sqrt(3). - Paul Barry (pbarry(AT)wit.ie), Jan 28 2004

a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k} - Paul Barry (pbarry(AT)wit.ie), Jul 28 2004

Euler transform of length 6 sequence [1, -1, -1, 0, 0, 1]. - Michael Somos Sep 23 2005

a(n)=-(1/6)*{n mod 6+(n+1) mod 6-[(n+3) mod 6]-[(n+4) mod 6]} - Paolo P. Lava (ppl(AT)spl.at), Oct 20 2006

a(n)=Sum_{k, 0<=k<=n}(-2)^(n-k)*A085838(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2006

a(n)=b(n+1) where b(n) is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 6), b(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos Oct 29 2006

a(1-n)=a(n). a(-2-n)=-a(n). - Michael Somos Feb 14 2006

Given g.f. A(x), then, B(x)=x*A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v +2*u*v*(-1 +u) . /* Michael Somos Oct 29 2006 */

a(n)=Sum_{k, 0<=k<=n}A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2006

a(n)=Sum_{k, 0<=k<=n}A133607(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 30 2007

a(n)=sum{k=0..n, C(n+k+1,2k+1)*(-1)^k}. [From Paul Barry (pbarry(AT)wit.ie), Jun 03 2009]

with(numtheory, cyclotomic); c := series(1/cyclotomic(6, x), x, 102): seq(coeff(c, x, n), n=0..101);

a:=n->coeftayl(1/(x^2-x+1), x=0, n);

a:=n->2*sin(Pi*(n+1)/3)/sqrt(3);

A010892:=n->[1, 1, 0, -1, -1, 0][irem(n, 6)+1];

A010892:=n->Array(0..5, [1, 1, 0, -1, -1, 0])[irem(n, 6)];

A010892:=n->table([0=1, 1=1, 2=0, 3=-1, 4=-1, 5=0])[irem(n, 6)];

with(numtheory, cyclotomic); c := series(1/cyclotomic(6, x), x, 102): seq(coeff(c, x, n), n=0..101); - Rainer Rosenthal (r.rosenthal(AT)web.de), Jan 01 2007

(PARI) a(n)=(-1)^(n\3)*sign((n+1)%3) /* Michael Somos Sep 23 2005 */

(PARI) a(n)=subst(poltchebi(n)+poltchebi(n-1), 'x, 1/2)*2/3 /* Michael Somos Sep 23 2005 */

(PARI) a(n)=[1, 1, 0, -1, -1, 0][n%6+1] /* Michael Somos Feb 14 2006

(PARI) {a(n)=local(A, p, e); if(n<0, 0, n++; A=factor(n); prod( k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, -(-1)^e, if(p==3, 0, if(p%6==1, 1, (-1)^e))))))} /* Michael Somos Oct 29 2006 */

# Python program from Alec Mihailovs, January 1 2007.

def A010892(n): return [1, 1, 0, -1, -1, 0][n%6]

(Other) sage: [lucas_number1(n, 1, +1) for n in xrange(-5, 97)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

a(2*n) = A057078(n), a(2*n+1) = A049347(n).

a(n) = row sums of signed triangle A049310.

Cf. A049347, A057078.

a(n)=A128834(n+1) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]

Sequence in context: A016350 A117441 A049347 this_sequence A091338 A016345 A016148

Adjacent sequences: A010889 A010890 A010891 this_sequence A010893 A010894 A010895

sign,easy,mult

Simon Plouffe (simon.plouffe(AT)gmail.com)

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jul 16 2004