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%I #18 Jul 03 2018 02:35:08
%S 2,8,72,450,3200,21632,149058,1019592,6993800,47922050,328499712,
%T 2251473408,15432082562,105772401800,724976569800,4969058770242,
%U 34058447431808,233440040239232,1600021920672450,10966713178192200
%N a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.
%C The relevant graph has rotational symmetry so the number of tilings is a square or twice a square, in this case by the formula for a(n) it is always twice a square.
%H Harry J. Smith, <a href="/A060635/b060635.txt">Table of n, a(n) for n = 1..200</a>
%H M. Ciucu, <a href="https://dx.doi.org/10.1006/jcta.1996.2725">Enumeration of perfect matchings in graphs with reflective symmetry</a>, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
%H W. Jockusch, <a href="https://dx.doi.org/10.1016/0097-3165(94)90006-X">Perfect matchings and perfect squares</a> J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F a(n) = 2 * F(n)^2 * F(n+1)^2 where F(n) is the n-th Fibonacci number - sequence A000045.
%F G.f.: -2*x*(1-x+x^2) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - _R. J. Mathar_, Jan 30 2011
%F a(n) = -4*(-1)^n*A002878(n)/25 - 2/25 + 6*A049658(n)/25. - _R. J. Mathar_, Jan 30 2011
%e a(1) = 2 because in this case the set S is the unit square and there is one horizontal tiling and one vertical.
%p with(combinat): for n from 1 to 40 do printf(`%d,`,2*fibonacci(n)^2*fibonacci(n+1)^2) od:
%o (PARI) { a=1; b=0; c=1; for (n=1, 200, f=a+b; g=b+c; a=b; b=c; c=g; write("b060635.txt", n, " ", 2*f^2*g^2); ) } \\ _Harry J. Smith_, Jul 08 2009
%Y Cf. A001654, A006253, A004003, A006125.
%K nonn
%O 1,1
%A Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 16 2001
%E More terms from _James A. Sellers_, Apr 16 2001
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