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A010892
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Inverse of 6th cyclotomic polynomial. A period 6 sequence.
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92
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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
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REFERENCES
| Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Ralph E. Griswold, Shaft Sequences
Index entries for sequences related to Chebyshev polynomials.
M. Somos, Rational Function Multiplicative Coefficients
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FORMULA
| 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 (paoloplava(AT)gmail.com), 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(2*n) = A057078(n), a(2*n+1) = A049347(n).
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]
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MAPLE
| 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
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PROG
| (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]
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CROSSREFS
| 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: A117441 A049347 * A091338 A016345 A016148 A016333
Adjacent sequences: A010889 A010890 A010891 * A010893 A010894 A010895
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KEYWORD
| sign,easy,mult
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AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jul 16 2004
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