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A130151
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Hexaperiodic sequence: repeat 1 1 1 -1 -1 -1 .
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9
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1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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LINKS
| M. Somos, Rational Function Multiplicative Coefficients
Index to sequences with linear recurrences with constant coefficients, signature (0,0,-1).
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FORMULA
| a(0)=a(1)=a(2)=-a(3)=-a(4)=-a(5)=1, a(n+6)=a(n) n=0,1.. .
a(n)=(1/3)*{-(n mod 6)+[(n+3) mod 6]}, with n>=0. - Paolo P. Lava (paoloplava(AT)gmail.com), Aug 28 2007
a(n) = ((-1)^n * (4 * (cos((2*n + 1)*Pi/3) + cos(n*Pi)) + 1) - 4) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 01 2007
a(n) = (-1)^n * (4 * cos((2*n + 1) * Pi/3) + 1) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 02 2007
G.f.: (1+x+x^2)/(1+x)/(x^2-x+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 14 2007
a(n)=3a(n-1)-a(n-3)+3a(n-4). - Paul Curtz (bpcrtz(AT)free.fr), Nov 22 2007
Closed form: a(n)=(1/3)*[1/2-(1/2*I)*sqrt(3)]^n+(1/3)*(-1)^n+(1/3)*[1/2+(1/2*I)*sqrt(3)]^n+[(1/3)*I]*{1/2 -[(1/2)*I]*sqrt(3)}^n*sqrt(3)-[(1/3)*I]*{1/2+[(1/2)*I]*sqrt(3)}^n*sqrt(3), with n>=0 and I=sqrt(-1) - Paolo P. Lava (paoloplava(AT)gmail.com), Jul 17 2008
a(n) = (-1)^floor(n/3). Compare with A057077, A143621 and A143622. Define E(k) = sum {n = 0..inf} a(n)*n^k/n! for k = 0,1,2,... . Then E(k) is an integral linear combination of E(0), E(1) and E(2) (a Dobinski-type relation). Precisely, E(k) = A143628(k) *E(0) + A143629(k) *E(1) + A143630(k) *E(2). [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]
Euler transform of length 6 sequence [ 1, 0, -2, 0, 0, 1]. - Michael Somos Feb 26 2011
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = -(-1)^e if e>0, b(p^e) = 1 if p == 1 (mod 4), b(p^e) = (-1)^e if p == 3 (mod 4) and p>3. - Michael Somos Feb 26 2011
a(n + 3) = a(-1 - n) = -a(n). - Michael Somos Feb 26 2011
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EXAMPLE
| 1 + x + x^2 - x^3 - x^4 - x^5 + x^6 + x^7 + x^8 - x^9 - x^10 - x^11 + ...
q + q^3 + q^5 - q^7 - q^9 - q^11 + q^13 + q^15 + q^17 - q^19 - q^21 + ...
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PROG
| (PARI) {a(n) = (-1) ^ (n\3)} /* Michael Somos Feb 26 2011 */
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CROSSREFS
| Cf. A131561, A131531.
A057077, A143621, A143622, A143628, A143629, A143630. [From Peter Bala (pbala(AT)toucansurf.com), Aug 28 2008]
Sequence in context: A112865 A114523 * A143431 A064179 A065357 A121241
Adjacent sequences: A130148 A130149 A130150 * A130152 A130153 A130154
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KEYWORD
| sign,easy
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Aug 03 2007
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