OFFSET
1,2
COMMENTS
This is a variation on A102620.
LINKS
Sébastien Palcoux, Is this representation of Go (game) irreducible? (version: 2019-09-22), MathOverflow.
FORMULA
a(n)/A102620(n) converges to 1.44066.... This would imply that a(n+1)/a(n) converges to 2.769292354... the real root of x^3 - 3*x^2 + x - 1 = 0.
From Colin Barker, Sep 26 2019: (Start)
G.f.: x*(1 + x + 3*x^2 - x^3) / ((1 - x)*(1 - 3*x + x^2 - x^3)).
a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-3) - a(n-4) for n > 4.
(End)
From Zhujun Zhang, Sep 28 2020: (Start)
a(n) = r_1^n + r_2^n + r_3^n - 2 where r_1, r_2 and r_3 are roots of x^3 - 3*x^2 + x - 1 = 0 for n > 0.
a(n) = floor(r^n - 3/2) where r is the real root of x^3 - 3*x^2 + x - 1 = 0 for n > 2.
(End)
EXAMPLE
PROG
(SageMath)
cpdef GoCycle(int n):
cdef int i, j, a, l
cdef list L, LL, T
LL=[]
for i in range(3**n):
L=Integer(i).digits(base=3, padto=n)
T=[L[0]]
for j in range(n-1):
if L[j+1]<>L[j]:
T.append(L[j+1])
if len(T)>1 and T[0]==T[-1]:
T.pop(0)
a=1
if 1 in T:
a=0
l=len(T)
if l>2:
for j in range(-2, l-2):
if not 1 in [T[j], T[j+1], T[j+2]]:
a=1
break
if a==0:
L=[j-1 for j in L]
LL.append(L)
return LL
[len(GoCycle(i)) for i in range(1, 17)]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Sébastien Palcoux, Sep 26 2019
STATUS
approved