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A000748
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Expansion of bracket function.
(Formerly M2520 N0995)
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12
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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; internal format)
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OFFSET
| 0,2
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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((-1)^j*binomial(n,3*j+1),j=0..floor(n/3))|. - John W. Layman, Sep 05 2003
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REFERENCES
| H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260.
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).
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| G.f.: 1/((1+x)^3-x^3).
a(n) = A007653(3^n).
a(n) = -3a(n-1)-3a(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
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MAPLE
| A000748:=(-1-2*z-3*z**2-3*z**3+18*z**5)/(-1+z+9*z**5); [Conjectured by S. 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..37); # Alois P. Heinz, Sep 06 2008
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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 */
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CROSSREFS
| Cf. A000749, A000750, A001659.
Cf. A057682.
Sequence in context: A021077 A114041 A057083 * A198373 A160178 A011383
Adjacent sequences: A000745 A000746 A000747 * A000749 A000750 A000751
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KEYWORD
| sign,easy,eigen
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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