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 A331235 The number of simple polygons having all points of a 3 X n grid as vertices. 0
 0, 1, 8, 62, 532, 4846, 45712, 441458, 4337468, 43187630, 434602280, 4411598154, 45107210436, 464047175770, 4799184825632, 49860914628042, 520109726201420, 5444641096394926, 57176049036449464, 602125661090565914, 6357215467283967404, 67274331104623532042 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The polygons are allowed to have flat angles (angles of exactly Pi) at some of the grid points. Empirically this sequence appears to be asymptotic to phi^(5n)/(66n), where phi is the golden ratio. LINKS David Eppstein, Counting grid polygonalizations, Jan 12 2020 PROG (Python) from math import log memo = {} def K(x, y, z):     """Number of strings of length y from two sorted alphabets of lengths x, z"""     if (x, y, z) in memo:         return memo[(x, y, z)]     if y == 0:         result = 1     else:         # i = length of the last block of equal characters in the string         # xx or zz = the largest remaining character in its alphabet         result = (sum(K(xx, y-i, z) for xx in range(x) for i in range(1, y+1)) +                  sum(K(x, y-i, zz) for zz in range(z) for i in range(1, y+1)))     memo[(x, y, z)] = result     return result def GC(n):     """Number of polygonalizations of 3xn grid"""     sum = 0     for i in range(n-1):    # number of points in K(...) can be up to n-2         mid = K(n-1, i, n-1)         for left in range(n-1-i):             right = n-2-i-left             contrib = mid             if left:                 contrib *= 2             if right:                 contrib *= 2             sum += contrib     return sum def exponent(p):     return p**(-4*p) * (1-p)**(-2*(1-p)) * (1-2*p)**(-1*(1-2*p)) base = ( (1+5**0.5)/2 )**5 #for n in range(2, 50): #    print(n, (base**n/(66*n))/GC(n), GC(n)) [GC(n) for n in range(1, 50)] CROSSREFS Sequence in context: A190975 A287815 A244831 * A053095 A099335 A194930 Adjacent sequences:  A331232 A331233 A331234 * A331236 A331237 A331238 KEYWORD nonn,easy AUTHOR David Eppstein, Jan 12 2020 STATUS approved

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Last modified January 19 16:16 EST 2021. Contains 340270 sequences. (Running on oeis4.)