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A327821 Number of legal Go positions on a board which is an n-cycle graph. 1
1, 5, 19, 57, 161, 449, 1247, 3457, 9577, 26525, 73459, 203433, 563369, 1560137, 4320479, 11964673 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is a variation on A102620.

LINKS

Table of n, a(n) for n=1..16.

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

A 2-cycle is a 1 X 2 grid so that a(2) = A102620(2) = A266278(1) = 5.

A 4-cycle is a 2 X 2 grid so that a(4) = A094777(2) = A266278(2) = 57.

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

Cf. A094777, A102620, A266278, A268113.

Sequence in context: A295776 A202658 A202118 * A201082 A109415 A029861

Adjacent sequences:  A327818 A327819 A327820 * A327822 A327823 A327824

KEYWORD

nonn,more

AUTHOR

Sébastien Palcoux, Sep 26 2019

STATUS

approved

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Last modified September 30 09:12 EDT 2022. Contains 357104 sequences. (Running on oeis4.)