Search: id:A010892
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%I A010892
%S A010892 1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,
%T A010892 0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,
%U A010892 1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0
%V A010892 1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,
               1,1,0,-1,-1,0,1,
%W A010892 1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,
               1,0,-1,-1,0,1,1,
%X A010892 0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0
%N A010892 Inverse of 6th cyclotomic polynomial. A period 6 sequence.
%C A010892 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).
%C A010892 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
%C A010892 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
%D A010892 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A010892 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A010892 Ralph E. Griswold, <a href="http://www.cs.arizona.edu/patterns/sequences.html">
               Shaft Sequences</a>
%H A010892 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A010892 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.
%F A010892 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).
%F A010892 a(n) = sqrt(4/3)*Imaginary[(1/2+i*sqrt(3/4))^(n+1)] - Henry Bottomley 
               (se16(AT)btinternet.com), Apr 12 2000
%F A010892 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
%F A010892 a(n)=2sin(pi*n/3+pi/3)/sqrt(3). - Paul Barry (pbarry(AT)wit.ie), Jan 
               28 2004
%F A010892 a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k} - Paul Barry (pbarry(AT)wit.ie), 
               Jul 28 2004
%F A010892 Euler transform of length 6 sequence [1, -1, -1, 0, 0, 1]. - Michael 
               Somos Sep 23 2005
%F A010892 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
%F A010892 a(n)=Sum_{k, 0<=k<=n}(-2)^(n-k)*A085838(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 26 2006
%F A010892 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
%F A010892 a(1-n)=a(n). a(-2-n)=-a(n). - Michael Somos Feb 14 2006
%F A010892 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 */
%F A010892 a(n)=Sum_{k, 0<=k<=n}A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 14 2006
%F A010892 a(n)=Sum_{k, 0<=k<=n}A133607(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 30 2007
%F A010892 a(n)=sum{k=0..n, C(n+k+1,2k+1)*(-1)^k}. [From Paul Barry (pbarry(AT)wit.ie), 
               Jun 03 2009]
%p A010892 with(numtheory,cyclotomic); c := series(1/cyclotomic(6,x),x,102): seq(coeff(c,
               x,n),n=0..101);
%p A010892 a:=n->coeftayl(1/(x^2-x+1), x=0, n);
%p A010892 a:=n->2*sin(Pi*(n+1)/3)/sqrt(3);
%p A010892 A010892:=n->[1,1,0,-1,-1,0][irem(n,6)+1];
%p A010892 A010892:=n->Array(0..5,[1,1,0,-1,-1,0])[irem(n,6)];
%p A010892 A010892:=n->table([0=1,1=1,2=0,3=-1,4=-1,5=0])[irem(n,6)];
%p A010892 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
%o A010892 (PARI) a(n)=(-1)^(n\3)*sign((n+1)%3) /* Michael Somos Sep 23 2005 */
%o A010892 (PARI) a(n)=subst(poltchebi(n)+poltchebi(n-1),'x,1/2)*2/3 /* Michael 
               Somos Sep 23 2005 */
%o A010892 (PARI) a(n)=[1,1,0,-1,-1,0][n%6+1] /* Michael Somos Feb 14 2006
%o A010892 (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 */
%o A010892 # Python program from Alec Mihailovs, January 1 2007.
%o A010892 def A010892(n): return [1,1,0,-1,-1,0][n%6]
%o A010892 (Other) sage: [lucas_number1(n,1,+1) for n in xrange(-5, 97)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A010892 a(2*n) = A057078(n), a(2*n+1) = A049347(n).
%Y A010892 a(n) = row sums of signed triangle A049310.
%Y A010892 Cf. A049347, A057078.
%Y A010892 a(n)=A128834(n+1) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Dec 05 2008]
%Y A010892 Sequence in context: A016350 A117441 A049347 this_sequence A091338 A016345 
               A016148
%Y A010892 Adjacent sequences: A010889 A010890 A010891 this_sequence A010893 A010894 
               A010895
%K A010892 sign,easy,mult
%O A010892 0,1
%A A010892 Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A010892 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jul 16 2004

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