OFFSET
0,4
COMMENTS
Derived turbulence series from A235347.
LINKS
Fung Lam, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: ( exp(4*Pi*i/3)*u + exp(2*Pi*i/3)*v - 1/6 )/x, where i=sqrt(-1),
u = 1/6*(-10-63*x+3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3), and
v = 1/6*(-10-63*x-3*sqrt(-24*x^3+357*x^2+42*x-27))^(1/3).
a(n) ~ sqrt((1-s)^3 / (2*s*(3 - 3*s + s^2))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where s = 0.31472177038151893868... is the root of the equation 1-2*s-5*s^2+4*s^3 = 0, and r = s*(1-s-2*s^2)/(1-s) = 0.22374229727550306625... - Vaclav Kotesovec, Jan 23 2014
D-finite with recurrence 117*n*(n-1)*a(n) -7*(n-1)*(35*n-66)*a(n-1) +21*(-69*n^2+269*n-254)*a(n-2) +(937*n^2-6403*n+10920)*a(n-3) -28*(n-4)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023
MATHEMATICA
CoefficientList[InverseSeries[Series[x*(1-x-2*x^2)/(1-x), {x, 0, 20}], x], x] (* Vaclav Kotesovec, Jan 22 2014 *)
PROG
(Python)
a = [0, 1]
for n in range(20):
m = len(a)
d = 0
for i in range (1, m):
for j in range (1, m):
if (i+j)%m == 0 and (i+j) <= m:
d += a[i]*a[j]
g = 0
for i in range (1, m-1):
for j in range (1, m-1):
for k in range (1, m-1):
if (i+j+k)%m == 0 and (i+j+k) <= m:
g += a[i]*a[j]*a[k]
y = 2*g + d - a[m-1]
a.append(y)
print(a)
(PARI) Vec(serreverse(x*(1-x-2*x^2)/(1-x)+O(x^66))) \\ Joerg Arndt, Jan 17 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Fung Lam, Jan 16 2014
EXTENSIONS
Prepended a(0)=0 to adapt to offset 0, Joerg Arndt, Jan 23 2014
b-file shifted for offset 0, Vaclav Kotesovec, Jan 23 2014
STATUS
approved