A130151 - OEIS

%I #41 Dec 14 2023 05:26:58

%S 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,

%T -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,

%U 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

%N Period 6: repeat [1, 1, 1, -1, -1, -1].

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1).

%F a(n+6) = a(n), a(0)=a(1)=a(2)=-a(3)=-a(4)=-a(5)=1.

%F 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

%F a(n) = (-1)^n * (4 * cos((2*n + 1) * Pi/3) + 1) / 3. - Federico Acha Neckar (f0383864(AT)hotmail.com), Sep 02 2007

%F G.f.: (1+x+x^2)/((1+x)*(x^2-x+1)). - _R. J. Mathar_, Nov 14 2007

%F a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4) for n>3. - _Paul Curtz_, Nov 22 2007

%F a(n) = (-1)^floor(n/3). Compare with A057077, A143621 and A143622. Define E(k) = Sum_{n >= 0} 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). - _Peter Bala_, Aug 28 2008

%F Euler transform of length 6 sequence [1, 0, -2, 0, 0, 1]. - _Michael Somos_, Feb 26 2011

%F 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

%F a(n + 3) = a(-1 - n) = -a(n) for all n in Z. - _Michael Somos_, Feb 26 2011

%F a(n) = (-1)^n * A257075(n) for all n in Z. - _Michael Somos_, Apr 15 2015

%F G.f.: 1 / (1 - x / (1 + 2*x^2 / (1 + x / (1 + x / (1 - x))))). - _Michael Somos_, Apr 15 2015

%F From _Wesley Ivan Hurt_, Jul 05 2016: (Start)

%F a(n) + a(n-3) = 0 for n>2.

%F a(n) = (cos(n*Pi) + 2*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3)) / 3. (End)

%F a(n)*a(n-4) = a(n-1)*a(n-3) for all n in Z. - _Michael Somos_, Feb 25 2020

%e G.f. = 1 + x + x^2 - x^3 - x^4 - x^5 + x^6 + x^7 + x^8 - x^9 - x^10 - x^11 + ...

%e G.f. = q + q^3 + q^5 - q^7 - q^9 - q^11 + q^13 + q^15 + q^17 - q^19 - q^21 + ...

%p seq(op([1, 1, 1, -1, -1, -1]), n=0..30); # _Wesley Ivan Hurt_, Jul 05 2016

%t a[ n_] := (-1)^Quotient[n, 3]; (* _Michael Somos_, Apr 24 2014 *)

%t PadRight[{}, 100, {1, 1, 1, -1, -1, -1}] (* _Wesley Ivan Hurt_, Jul 05 2016 *)

%o (PARI) {a(n) = (-1) ^ (n\3)}; /* _Michael Somos_, Feb 26 2011 */

%o (Magma) &cat [[1, 1, 1, -1, -1, -1]^^20]; // _Wesley Ivan Hurt_, Jul 05 2016

%Y Cf. A131561, A131531, A257075.

%Y Cf. A057077, A143621, A143622, A143628, A143629, A143630. - _Peter Bala_, Aug 28 2008

%K sign,easy

%O 0,1

%A _Paul Curtz_, Aug 03 2007