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 A084103 Expansion of (1+x)^3/(1+x^3). 6
 1, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3, 0, 3, 3, 0, -3, -3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Partial sums are A084104. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1,-1). FORMULA a(n) = Sum_{k=0..n} binomial(2n-k-1, k)(-1)^k*3(n-k). - Paul Barry, Jan 21 2005 a(0) = 1 and a(n) = 2*sqrt(3)*sin(n*Pi/3). - N-E. Fahssi, Mar 04 2010 Euler transform of length 6 sequence [3, -3, -1, 0, 0, 1]. - Michael Somos, Feb 13 2011 a(n) = -a(-n) = 3 * A128834(n) except a(0) = 1. - Michael Somos, Feb 13 2011 a(n) = 3*(n^2 mod 3)*(-1)^floor(n/3), n>0. - Wesley Ivan Hurt, May 15 2015 The periodic sequence b(n) = a(n+1) has the o.g.f. 3 + G(x) = 3 + 3x(1-x) / (1-x(1-x)) = 3 + 3 L(Cinv(x)) = 3 + 3 x - 3 x^3 - 3 x^4 + ... , where L(x) = x/(1-x) with inverse Linv(x) = x/(1+x) and Cinv(x) = x(1-x), the inverse of the o.g.f. for the shifted Catalan numbers of A000108, C(x) = (1-sqrt(1-4x))/2. Then Ginv(x) = C(Linv(x/3)) = [1 - sqrt[1-4x/(3+x)]]/2. Cf. A267633. - Tom Copeland, Jan 25 2016 EXAMPLE 1 + 3*x + 3*x^2 - 3*x^4 - 3*x^5 + 3*x^7 + 3*x^8 - 3*x^10 - 3*x^11 + ... MAPLE 1, seq(op((-1)^i*[3, 3, 0]), i=0..20); # Robert Israel, May 17 2015 MATHEMATICA CoefficientList[Series[(1 + x)^3/(1 + x^3), {x, 0, 100}], x] (* Vincenzo Librandi, May 16 2015 *) Join[{1}, LinearRecurrence[{1, -1}, {3, 3}, 30]] (* G. C. Greubel, Jan 15 2018 *) PROG (PARI) {a(n) = (n==0) + [0, 3, 3, 0, -3, -3][n%6 + 1]} /* Michael Somos, Feb 13 2011 */ (PARI) {a(n) = (n==0) - 3 * (-1)^n * kronecker(-3, n)} /* Michael Somos, Feb 13 2011 */ (Magma) I:=[1, 3, 3]; [n le 3 select I[n] else Self(n-1)-Self(n-2): n in [1..100]]; // Vincenzo Librandi, May 16 2015 CROSSREFS Cf. A000108, A267633. Sequence in context: A098316 A160165 A084055 * A036477 A330013 A128164 Adjacent sequences: A084100 A084101 A084102 * A084104 A084105 A084106 KEYWORD easy,sign AUTHOR Paul Barry, May 15 2003 STATUS approved

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Last modified March 21 14:15 EDT 2023. Contains 361408 sequences. (Running on oeis4.)