cp's OEIS Frontend

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

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, Nov 14 2007

a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4) for n>3. - Paul Curtz, Nov 22 2007

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

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) for all n in Z. - Michael Somos, Feb 26 2011

a(n) = (-1)^n * A257075(n) for all n in Z. - Michael Somos, Apr 15 2015

G.f.: 1 / (1 - x / (1 + 2*x^2 / (1 + x / (1 + x / (1 - x))))). - Michael Somos, Apr 15 2015

From Wesley Ivan Hurt, Jul 05 2016: (Start)

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

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

a(n)*a(n-4) = a(n-1)*a(n-3) for all n in Z. - Michael Somos, Feb 25 2020

Euler transform of length 6 sequence [-1, 1, 2, 0, 0, -1].

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -1 if e>0, otherwise b(p^e) = 1.

a(n) = a(-1-n) = a(n+3) = -a(n-1)*a(n-2) for all n in Z.

G.f.: (1 - x + x^2) / (1 - x^3).

G.f.: (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^3)^2).

G.f.: 1 / (1 + x / (1 + 2*x^2 / (1 - x / (1 - x / (1 + x))))).

Given g.f. A(x), then x*A(x^2) = Sum_{k>0} (x^k - x^(2*k)) - 2*(x^(3*k) - x^(6*k)).

a(n) = A131561(n+1) for all n in Z.

a(n) = (-1)^n * A130151(n) for all n in Z.

Convolution inverse is A257076.

PSUM transform is A008611.

BINOMIAL transform is A086953.

1 / (1 - a(0)*x / (1 - a(1)*x / (1 - a(2)*x / ...))) is the g.f. of A168505.

From Wesley Ivan Hurt, Jul 02 2016: (Start)

a(n) = (1 + 2*cos(2*n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/3.

a(n) = 2*sgn((n+2) mod 3) - 1. (End)

E.g.f.: (exp(3*x/2) + 4*sin(Pi/6-sqrt(3)*x/2))*exp(-x/2)/3. - Ilya Gutkovskiy, Jul 02 2016

From R. J. Mathar, Oct 23 2010: (Start)

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7).

G.f.: -x*(1+2*x+2*x^3+x^4-3*x^2) / ( (1+x+x^2)^2*(x-1)^3 ).

a(n) = 2*A061347(n+1)/9 +4/9 + n*(n+1)/6 + 2*b(n)/3 where b(3k+1) = 0, b(3k) = -3k - 1 and b(3k+2) = 3k + 3. (End)

a(n) = sum((i+1)*A131561(i), i=0..n-1) = A000217(n)-6*A000217(floor(n/3)). [Bruno Berselli, Dec 10 2010]

a(0) = 0, a(n) = a(n-1) + (-1)^((n + 1) mod 3)*n - Jon Perry, Feb 17 2013

a(n) = n*(n+1)/2-3*floor(n/3)*(floor(n/3)+1). - R. J. Cano, Mar 01 2013 [Same as Berselli's formula. - Ed.]

a(3k) = 3k(k-1)/2. - Jon Perry, Mar 01 2013

a(0) = 0, a(n) = a(n-1) + (1 - ((n+1) mod 3 mod 2) * 2) * n. - Jon Perry, Mar 03 2013

G.f.: 1/(1-x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+... (continued fraction, defined by the sequences (1,0,0,2,0,0,2,0,0,2,0,...) and (-1,1,-1,-1,1,-1,...));

g.f.: (1+x-sqrt(1-2x+x^2+4x^3))/(2x(1-x)).

a(n) = Sum_{k=0..n} A198379(n,k)*(-1)^(n-k). - Philippe Deléham, Oct 29 2011

a(n) = (-1)^n*Sum_{k=0..n} A174014(n,k)*(-2)^k. - Philippe Deléham, Feb 16 2012

G.f.: (1+x)/(G(0)+x), where G(k) = 1 - x + x^3/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 29 2013

Conjecture: (n+1)*a(n) - 3*n*a(n-1) + 3*(n-1)*a(n-2) + 3*(n-4)*a(n-3) + 2*(-2*n+7)*a(n-4) = 0. - R. J. Mathar, Feb 10 2015

G.f. A(x) satisfies (A(x) - 1) / A(x)^2 = (x - x^2) / (1 + x). - Michael Somos, Jan 20 2017

0 = a(n)*(+16*a(n+1) - 6*a(n+2) - 42*a(n+3) + 54*a(n+4) - 22*a(n+5)) + a(n+1)*(-18*a(n+1) + 27*a(n+2) + 6*a(n+3) - 31*a(n+4) + 18*a(n+5))+ a(n+2)*(-18*a(n+2) + 36*a(n+3) - 30*a(n+4) + 9*a(n+5)) + a(n+3)*(+6*a(n+4) - 6*a(n+5)) + a(n+4)*(+a(n+5)) if n >= 0. - Michael Somos, Jan 20 2017

3 - sqrt(5) = Sum_{n>=1} 1/(F(2*n)*F(2*n+1)) + 1/(F(2*n)*F(2*n+2)) - 1/(F(2*n+1)*F(2*n+2)), where F = A000045 (Fibonacci numbers).

From Colin Barker, Dec 17 2015: (Start)

a(n) = 8*a(n-3) - 8*a(n-6) + a(n-9) for n>9.

G.f.: x*(2+3*x+6*x^2-x^3-8*x^5+x^8) / ((1-x)*(1+x+x^2)*(1-7*x^3+x^6)).

(End)

a(n) = A261327(n) + A131561(n+2) = (n^2 + 4)*(5 - 3*(-1)^n)/8 + (-1)^((n+1) mod 3).

From Colin Barker, Nov 24 2019: (Start)

G.f.: 3*x*(2 + 3*x + x^2 - 2*x^3 + x^4 + 3*x^5 + 2*x^6) / ((1 - x)^3*(1 + x)^3*(1 + x + x^2)).

a(n) = -a(n-1) + 2*a(n-2) + 3*a(n-3) - 3*a(n-5) - 2*a(n-6) + a(n-7) + a(n-8) for n>8. (End)

a(n) = 2*a(n-3) - a(n-6) for n > 5.

a(n)= (9 + 12*n - 6*n*A061347(n) + 6*(2 + 3*n)*A049347(n+2))/9.

G.f.: (1 + 5*x - 3*x^2 + 7*x^3 + 3*x^4 - x^5)/(1 - x^3)^2.

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(3 + 4*x) + 12*x*cos(sqrt(3)*x/2) + 4*sqrt(3)*(2 - 3*x)*sin(sqrt(3)*x/2))/9.

Sum_{n>=0} 1/a(n) = (log(2) + Pi)/4 = A377227.

a(n) = A130823(n-1) * A131534(n).

a(n) = A226023(n) - A226023(n-1) with A226023(-1)=-1.

a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7). [Ralf Stephan]

From Bruno Berselli, May 29 2013: (Start)

G.f.: (1+x)^3*(1-x+x^2)/((1-x)^2*(1+x+x^2)^2).

a(n) = 2*a(n-3)-a(n-6).

a(3n)*a(3n-1)-a(3n-2) = A016754(n-1), n>0. (End)