A060635 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.

2, 8, 72, 450, 3200, 21632, 149058, 1019592, 6993800, 47922050, 328499712, 2251473408, 15432082562, 105772401800, 724976569800, 4969058770242, 34058447431808, 233440040239232, 1600021920672450, 10966713178192200, 75166970919070472, 515202081704384258, 3531247605071972352

Offset: 1

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 16 2001

Keywords

• nonn

Comments

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.

Examples

			a(1) = 2 because in this case the set S is the unit square and there is one horizontal tiling and one vertical.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..200
• M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
• W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Crossrefs

Cf. A000045, A001654, A006253, A004003, A006125, A002878, A049658, A197424.

Programs

Maple

with(combinat): for n from 1 to 40 do printf(`%d,`,2*fibonacci(n)^2*fibonacci(n+1)^2) od:

Mathematica

2*Times @@@ Partition[Fibonacci[Range[25]]^2, 2, 1] (* Paolo Xausa, Jul 03 2025 *)

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

Formula

a(n) = 2 * F(n)^2 * F(n+1)^2 where F(n) is the n-th Fibonacci number - sequence A000045.

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

a(n) = -4*(-1)^n*A002878(n)/25 - 2/25 + 6*A049658(n)/25. - R. J. Mathar, Jan 30 2011

a(n) = 2 * A001654(n)^2 = 2 * A197424(n-2) for n>=2. - Alois P. Heinz, Jul 03 2025

Extensions

More terms from James Sellers, Apr 16 2001