A249582 Norm of the complex coefficients in 1/(1 - x + (1-2*I)*x^2).

1, 1, 4, 17, 29, 160, 377, 1377, 4468, 13369, 46573, 141440, 469169, 1499329, 4795556, 15600033, 49731901, 161026720, 516993193, 1663865633, 5361647252, 17231870281, 55519546637, 178586104320, 574860647521, 1850350458241, 5954177494084, 19166631789617, 61680287845469

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..28.

Formula

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

Example

G.f.: A(x) = 1 + x + 4*x^2 + 17*x^3 + 29*x^4 + 160*x^5 + 377*x^6 +...
From the complex series expansion:
1/(1 - x + (1-2*I)*x^2) = 1 + x + 2*I*x^2 + (-1 + 4*I)*x^3 + (-5 + 2*I)*x^4 +
(-12 - 4*I)*x^5 + (-11 - 16*I)*x^6 + (9 - 36*I)*x^7 + (52 - 42*I)*x^8 +
(115 + 12*I)*x^9 + (147 + 158*I)*x^10 +...
we obtain this sequence as the norm of the above coefficients:
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 2^2 = 4;
a(3) = (-1)^2 + 4^2 = 17;
a(4) = (-5)^2 + 2^2 = 29;
a(5) = (-12)^2 + (-4)^2 = 160; ...

Prog

(PARI) {a(n)=norm(polcoeff(1/(1-x+(1-2*I)*x^2 +x*O(x^n)), n))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=(polcoeff((1-5*x^2)/(1 - x - 8*x^2 - 5*x^3 + 25*x^4 +x*O(x^n)), n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A273073 A272330 A163736 * A127547 A302410 A303177

Adjacent sequences: A249579 A249580 A249581 * A249583 A249584 A249585

Keyword

nonn

Author

Paul D. Hanna, Nov 01 2014

Status

approved