A249582 - OEIS

%I #5 Nov 01 2014 21:42:03

%S 1,1,4,17,29,160,377,1377,4468,13369,46573,141440,469169,1499329,

%T 4795556,15600033,49731901,161026720,516993193,1663865633,5361647252,

%U 17231870281,55519546637,178586104320,574860647521,1850350458241,5954177494084,19166631789617,61680287845469

%N Norm of the complex coefficients in 1/(1 - x + (1-2*I)*x^2).

%C Limit a(n)^(1/n) = r = 0.3107068879... = (1+sqrt(73) - sqrt(2*sqrt(73)-6))/20 where r = norm(a+b*I) with a = (1 - sqrt((sqrt(73)-3)/2) + sqrt(2*sqrt(73)+6))/10 and b = (2 - sqrt(2*sqrt(73)-6) - sqrt((sqrt(73)+3)/2))/10 such that (a+b*I) is a root of 1 - x + (1-2*I)*x^2 = 0.

%F G.f.: (1-5*x^2)/(1 - x - 8*x^2 - 5*x^3 + 25*x^4).

%e G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 29*x^4 + 160*x^5 + 377*x^6 +...

%e From the complex series expansion:

%e 1/(1 - x + (1-2*I)*x^2) = 1 + x + 2*I*x^2 + (-1 + 4*I)*x^3 + (-5 + 2*I)*x^4 +

%e (-12 - 4*I)*x^5 + (-11 - 16*I)*x^6 + (9 - 36*I)*x^7 + (52 - 42*I)*x^8 +

%e (115 + 12*I)*x^9 + (147 + 158*I)*x^10 +...

%e we obtain this sequence as the norm of the above coefficients:

%e a(0) = 1^2 = 1;

%e a(1) = 1^2 = 1;

%e a(2) = 2^2 = 4;

%e a(3) = (-1)^2 + 4^2 = 17;

%e a(4) = (-5)^2 + 2^2 = 29;

%e a(5) = (-12)^2 + (-4)^2 = 160; ...

%o (PARI) {a(n)=norm(polcoeff(1/(1-x+(1-2*I)*x^2 +x*O(x^n)), n))}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=(polcoeff((1-5*x^2)/(1 - x - 8*x^2 - 5*x^3 + 25*x^4 +x*O(x^n)), n))}

%o for(n=0, 30, print1(a(n), ", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 01 2014