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Number of unrooted hexagonal polyominoes with n cells and no reflections allowed.
(Formerly M1750 N0694)
2

%I M1750 N0694 #27 Dec 29 2016 02:28:38

%S 1,1,2,7,18,64,226,856,3306,13249,53794,222717,933438,3960487,

%T 16970362,73381453,319817670,1403875491,6202064928,27559699507,

%U 123115236582,552654175124,2491870281372,11281732737898,51270697159708,233822055167579,1069835253304014,4909835353596645,22596879316320522

%N Number of unrooted hexagonal polyominoes with n cells and no reflections allowed.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002214/b002214.txt">Table of n, a(n) for n=1..200</a>

%H F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

%F G.f.: x + x*U(x) + (1/2)*(3*x-1)*U(x)^2 + (1/2)*(1+x)*U(x^2) + (1/3)*x*U(x)^3 + (2/3)*x*U(x^3), where U(x)=(1-3*x-sqrt(1-6*x+5*x^2))/(2*x). - _Emeric Deutsch_, May 02 2004

%F a(n) ~ 5^(n+1/2)/(2*sqrt(Pi)*n^(5/2)). - _Vaclav Kotesovec_, Aug 13 2013

%o (PARI) x = 'x+O('x^66);

%o U(x) = (1-3*x-sqrt(1-6*x+5*x^2))/(2*x);

%o gf = x + x*U(x) + (1/2)*(3*x-1)*U(x)^2 + (1/2)*(1+x)*U(x^2) + (1/3)*x*U(x)^3 + (2/3)*x*U(x^3);

%o Vec(gf) \\ _Joerg Arndt_, Aug 13 2013

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, May 02 2004