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%I #25 Apr 20 2014 13:11:26
%S 50,272,722,3108,10082,39952,140450,537636,1956242,7379216,27246962,
%T 102144036,379501250,1418981392,5285770562,19742287908,73621286642,
%U 274848860432,1025412242450,3827417932836,14282150107682,53304783436816,198924689265122,742414961433636,2770663499604050,10340361362903312
%N Number of perfect matchings in the graph C_4 X C_n.
%H Vincenzo Librandi, <a href="/A220864/b220864.txt">Table of n, a(n) for n = 3..1000</a>
%H S. Butler and S. Osborne, <a href="http://orion.math.iastate.edu/butler/papers/walk_tiling.pdf">Counting tilings by taking walks</a>, 2012.
%F G.f.: 2*x^3*(25+36*x-333*x^2-6*x^3+467*x^4-104*x^5-71*x^6+18*x^7)/((1-x)*(1+x)*(1-4*x+x^2)*(1-2*x-x^2)*(1+2*x-x^2)). - _Sergey Perepechko_, Oct 21 2013
%F Assuming the above o.g.f. we have, for n >= 1, a(2n+1) = 2*A001834(n)^2 = (2 + sqrt(3))^(2*n+1) + (2 - sqrt(3))^(2*n+1) - 2. - _Peter Bala_, Apr 19 2014
%t CoefficientList[Series[2 (25 + 36 x - 333 x^2 - 6 x^3 + 467 x^4 - 104 x^5 - 71 x^6 + 18 x^7)/((1 - x) (1 + x) (1 - 4 x + x^2) (1 - 2 x - x^2) (1 + 2 x - x^2)), {x,0, 40}], x] (* _Vincenzo Librandi_, Apr 20 2014 *)
%o (PARI) Vec( 2*x^3*(25+36*x-333*x^2-6*x^3+467*x^4-104*x^5-71*x^6+18*x^7)/((1-x)*(1+x)*(1-4*x+x^2)*(1-2*x-x^2)*(1+2*x-x^2)) +O(x^66) ) \\ _Joerg Arndt_, Oct 22 2013
%Y Cf. A001834.
%K nonn,easy
%O 3,1
%A _N. J. A. Sloane_, Dec 27 2012
%E More terms from _Joerg Arndt_, Oct 22 2013