A327821 Number of legal Go positions on a board which is an n-cycle graph.

1, 5, 19, 57, 161, 449, 1247, 3457, 9577, 26525, 73459, 203433, 563369, 1560137, 4320479, 11964673

Offset: 1

Author

Sébastien Palcoux, Sep 26 2019

Keywords

• nonn
• more

Comments

This is a variation on A102620.

Examples

			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.
		

Links

• Sébastien Palcoux, Is this representation of Go (game) irreducible? (version: 2019-09-22), MathOverflow.

Crossrefs

Cf. A094777, A102620, A266278, A268113.

Programs

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)]

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)