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 A000748 Expansion of bracket function. (Formerly M2520 N0995) 14
 1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729, -2187, 4374, -6561, 6561, 0, -19683, 59049, -118098, 177147, -177147, 0, 531441, -1594323, 3188646, -4782969, 4782969, 0, -14348907, 43046721, -86093442, 129140163, -129140163, 0, 387420489, -1162261467 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS It appears that the sequence coincides with its third-order absolute difference. - John W. Layman, Sep 05 2003 It appears that, for n > 0, the (unsigned) a(n) = 3*|A057682(n)| = 3*|Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1)|. - John W. Layman, Sep 05 2003 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 Index entries for linear recurrences with constant coefficients, signature (-3,-3). FORMULA G.f.: 1/((1+x)^3-x^3). a(n) = A007653(3^n). a(n) = -3*a(n-1) - 3*a(n-2). - Paul Curtz, May 12 2008 a(n) = -(1/2)*i*sqrt(3)*(-3/2 - (1/2)*i*sqrt(3))^n + (1/2)*i*sqrt(3)*(-3/2 + (1/2)*i*sqrt(3))^n + (1/2)*(-3/2 + (1/2)*i*sqrt(3))^n + (1/2)*(-3/2 - (1/2)*i*sqrt(3))^n, with n >= 0 and i=sqrt(-1). - Paolo P. Lava, Jun 11 2008 a(n) = Sum_{k=1..n} binomial(k,n-k)*(-3)^(k) for n > 0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011 G.f.: 1/(1 + 3*x /(1 - x /(1+x))). - Michael Somos, May 12 2012 G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - 3*x*(2*k+1 + x)/(3*x*(2*k+2 + x) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2014 a(n) = 2*3^(n/2)*sin((1-5*n)*Pi/6). - Jean-François Alcover, Mar 12 2014 EXAMPLE G.f. = 1 - 3*x + 6*x^2 - 9*x^3 + 9*x^4 - 27*x^6 + 81*x^7 - 162*x^8 + ... MAPLE A000748:=(-1-2*z-3*z**2-3*z**3+18*z**5)/(-1+z+9*z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from signs a:= n-> (Matrix([[ -3, 1], [ -3, 0]])^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Sep 06 2008 MATHEMATICA a[n_] := 2*3^(n/2)*Sin[(1-5*n)*Pi/6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2014 *) LinearRecurrence[{-3, -3}, {1, -3}, 40] (* Jean-François Alcover, Feb 11 2016 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff(1 / (1 + 3*x + 3*x^2) + x * O(x^n), n))}; /* Michael Somos, Jun 07 2005 */ (PARI) {a(n) = if( n<0, 0, 3^((n+1)\2) * (-1)^(n\6) * ((-1)^n + (n%3==2)))}; /* Michael Somos, Sep 29 2007 */ (Magma) I:=[1, -3]; [n le 2 select I[n] else -3*Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 11 2016 CROSSREFS Cf. A000749, A000750, A001659. Cf. A057682. Sequence in context: A325750 A325748 A057083 * A325738 A198373 A331065 Adjacent sequences: A000745 A000746 A000747 * A000749 A000750 A000751 KEYWORD sign,easy,eigen AUTHOR STATUS approved

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Last modified November 30 11:20 EST 2022. Contains 358441 sequences. (Running on oeis4.)