A272665 - OEIS

%I #19 Oct 06 2016 02:04:03

%S 0,0,1,2,4,6,7,4,-8,-36,-87,-162,-244,-278,-145,360,1520,3608,6641,

%T 9882,11028,5166,-16073,-64084,-149528,-272076,-399911,-436682,

%U -179684,712530,2698335,6192720,11140064,16170928,17258081,6043314,-31395292,-113477674

%N Imaginary parts of b(n) sequence used to define A143056.

%H Colin Barker, <a href="/A272665/b272665.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,-1).

%F From _Colin Barker_, May 17 2016: (Start)

%F a(n) = 2*a(n-1)-2*a(n-3)-a(n-4) for n>4.

%F G.f.: x^3 / (1-2*x+2*x^3+x^4).

%F (End)

%F a(n) = (sin((n-1)*theta)*(tau^(n-1) + (-tau)^(1-n))*phi^(3/2) - cos((n-1)*theta)*(tau^(n-1) - (-tau)^(1-n))/phi^(3/2))/(2*sqrt(5)), where phi=(1+sqrt(5))/2, tau=sqrt(phi+sqrt(phi)), theta=arctan(phi^(-3/2)). - _Vladimir Reshetnikov_, Oct 04 2016

%t Im[Fibonacci[Range[0, 20], 1 + I]] (* _Vladimir Reshetnikov_, Oct 04 2016 *)

%t LinearRecurrence[{2, 0, -2, -1}, {0, 0, 1, 2}, 36] (* _Robert G. Wilson v_, Oct 05 2016 *)

%o (PARI) concat(vector(2), Vec(x^3/(1-2*x+2*x^3+x^4) + O(x^50))) \\ _Colin Barker_, May 17 2016

%Y Cf. A143056.

%K sign,easy

%O 1,4

%A _N. J. A. Sloane_, May 17 2016