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

0,3

Comments

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).

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

Links

Table of n, a(n) for n=0..46.

Index entries for linear recurrences with constant coefficients, signature (-2,-2).

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

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

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

Example

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 + ...

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 *)

Prog

(PARI) x='x+O('x^66) /* that many terms */
Vec(serlaplace(cos(x)/exp(x))) /* show terms */
/* or */
vector(66, n, -real((-1+I)^n)) /* Joerg Arndt, May 13 2011 */
(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 */

Crossrefs

Cf. A009116.

Cf. A135353, A137444, A137426, A137429.

Sequence in context: A009116 A146559 A118434 * A199051 A300190 A099211

Adjacent sequences: A090129 A090130 A090131 * A090133 A090134 A090135

Keyword

easy,sign

Author

Paul Barry, Nov 21 2003

Status

approved