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%I #42 Apr 20 2019 03:36:44
%S 1,0,-2,4,-4,0,8,-16,16,0,-32,64,-64,0,128,-256,256,0,-512,1024,-1024,
%T 0,2048,-4096,4096,0,-8192,16384,-16384,0,32768,-65536,65536,0,
%U -131072,262144,-262144,0,524288,-1048576,1048576,0,-2097152,4194304,-4194304,0,8388608
%N Expansion of (1+2*x)/(1+2*x+2*x^2).
%C The expansion of (1-2x)/(1-2x+2x^2) has a(n) = Sum_{k=0..n} C(n,k)(-1)^(-k)(-1)^floor(k/2).
%C Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - _R. J. Mathar_, Aug 10 2012
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-2).
%F G.f.: (1+2*x)/(1+2*x+2*x^2).
%F a(n) = Sum_{k=0..n} C(n, k)*(-1)^(n-k)*(-1)^floor(k/2).
%F a(n) = sqrt(2)*2^(n/2)*sin(3*Pi*n/4+Pi/4). - _Paul Barry_, Feb 25 2004
%F a(n) = -a(n-1) + 2*a(n-3). - _Paul Curtz_, Apr 24 2008
%F Negated real part of (-1+i)^n, imaginary part is A108520. - _Joerg Arndt_, May 13 2011
%F E.g.f.: A(x)=(cos(x)+sin(x))/exp(x)=Q(0); 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). - _Sergei N. Gladkovskii_, Nov 28 2011
%F a(4*n + 1) = 0. a(2*n) = A120617(n). a(4*n + 3) = (-4)^n. - _Michael Somos_, May 25 2013
%F a(n) = - 2^(n-1)*a(2-n) for all n in Z. - _Michael Somos_, Jun 26 2017
%F a(n) = (I + 1)*((-1 - I)^n - I*(-1 + I)^n)/2. - _Taras Goy_, Apr 20 2019
%e 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 + ...
%t a[ n_] := Re[ -(I - 1)^(n + 1)]; (* _Michael Somos_, May 25 2013 *)
%t 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 *)
%t 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 *)
%t a[ n_] := Simplify[ -2 Sqrt[2]^(n - 1) ChebyshevT[ n + 1, -1 / Sqrt[2]]]; (* _Michael Somos_, May 25 2013 *)
%t LinearRecurrence[{-2,-2},{1,0},50] (* _Harvey P. Dale_, Oct 23 2017 *)
%o (PARI) x='x+O('x^66) /* that many terms */
%o Vec(serlaplace(cos(x)/exp(x))) /* show terms */
%o /* or */
%o vector(66, n, -real((-1+I)^n)) /* _Joerg Arndt_, May 13 2011 */
%o (PARI) {a(n) = real( -(I - 1)^(n + 1) )}; /* _Michael Somos_, May 25 2013 */
%o (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 */
%o (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 */
%o (PARI) {a(n) = simplify( -2 * quadgen(8)^(n - 1) * polchebyshev( n + 1, 1, -1 / quadgen(8)))}; /* _Michael Somos_, May 25 2013 */
%Y Cf. A009116.
%Y Cf. A135353, A137444, A137426, A137429.
%K easy,sign
%O 0,3
%A _Paul Barry_, Nov 21 2003
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