A090132 Expansion of (1+2*x)/(1+2*x+2*x^2).
1, 0, -2, 4, -4, 0, 8, -16, 16, 0, -32, 64, -64, 0, 128, -256, 256, 0, -512, 1024, -1024, 0, 2048, -4096, 4096, 0, -8192, 16384, -16384, 0, 32768, -65536, 65536, 0, -131072, 262144, -262144, 0, 524288, -1048576, 1048576, 0, -2097152, 4194304, -4194304, 0, 8388608
Offset: 0
Examples
G.f. = 1 - 2*x^2 + 4*x^3 - 4*x^4 + 8*x^6 - 16*x^7 + 16*x^8 - 32*x^10 + 64*x^11 + ...
Links
- Index entries for linear recurrences with constant coefficients, signature (-2,-2).
Programs
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Mathematica
a[ n_] := Re[ -(I - 1)^(n + 1)]; (* Michael Somos, May 25 2013 *) a[ n_] := If[ n < 0, - 2^(n-1) a[2 - n], SeriesCoefficient[ (1 + 2 x) / (1 + 2 x + 2 x^2), {x, 0, n}]]; (* Michael Somos, May 25 2013 *) a[ n_] := If[ n < 0, - 2^(n-1) a[2 - n], n! SeriesCoefficient[ (Cos[x] + Sin[x]) / Exp[x], {x, 0, n}]]; (* Michael Somos, May 25 2013 *) a[ n_] := Simplify[ -2 Sqrt[2]^(n - 1) ChebyshevT[ n + 1, -1 / Sqrt[2]]]; (* Michael Somos, May 25 2013 *) LinearRecurrence[{-2,-2},{1,0},50] (* Harvey P. Dale, Oct 23 2017 *)
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PARI
my(x='x+O('x^66)); Vec(serlaplace((cos(x)+sin(x))/exp(x))) \\ Joerg Arndt, May 13 2011 -
PARI
vector(66, n, -real((-1+I)^n)) /* Joerg Arndt, May 13 2011 */
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PARI
{a(n) = real( -(I - 1)^(n + 1) )}; /* Michael Somos, May 25 2013 */ -
PARI
{a(n) = if( n<0, - 2^(n-1) * a(2 - n), polcoeff( (1 + 2*x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))}; /* Michael Somos, May 25 2013 */ -
PARI
{a(n) = my(A); if( n<0, - 2^(n-1) * a(2 - n), A = x * O(x^n); n! * polcoeff( (cos(x + A) + sin(x + A)) / exp(x + A), n))}; /* Michael Somos, May 25 2013 */ -
PARI
{a(n) = simplify( -2 * quadgen(8)^(n - 1) * polchebyshev( n + 1, 1, -1 / quadgen(8)))}; /* Michael Somos, May 25 2013 */
Formula
G.f.: (1+2*x)/(1+2*x+2*x^2).
a(n) = Sum_{k=0..n} C(n, k)*(-1)^(n-k)*(-1)^floor(k/2).
a(n) = sqrt(2)*2^(n/2)*sin(3*Pi*n/4+Pi/4). - Paul Barry, Feb 25 2004
a(n) = -a(n-1) + 2*a(n-3). - Paul Curtz, Apr 24 2008
Negated real part of (-1+i)^n, imaginary part is A108520. - Joerg Arndt, May 13 2011
From Sergei N. Gladkovskii, Nov 28 2011: (Start)
E.g.f.: (cos(x) + sin(x))/exp(x).
E.g.f.: A(x) = Q(0), where Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). (End)
a(4*n + 1) = 0. a(2*n) = A120617(n). a(4*n + 3) = (-4)^n. - Michael Somos, May 25 2013
a(n) = - 2^(n-1)*a(2-n) for all n in Z. - Michael Somos, Jun 26 2017
a(n) = (I + 1)*((-1 - I)^n - I*(-1 + I)^n)/2. - Taras Goy, Apr 20 2019
Comments