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A060635
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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.
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1
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2, 8, 72, 450, 3200, 21632, 149058, 1019592, 6993800, 47922050, 328499712, 2251473408, 15432082562, 105772401800, 724976569800, 4969058770242, 34058447431808, 233440040239232, 1600021920672450, 10966713178192200
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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.
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REFERENCES
| 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
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LINKS
| Harry J. Smith, Table of n, a(n) for n=1,...,200
Index to sequences with linear recurrences with constant coefficients, signature (5,15,-15,-5,1).
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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
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EXAMPLE
| a(1) = 2 because in this case the set S is the unit square and there is one horizontal tiling and one vertical.
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MAPLE
| with(combinat): for n from 1 to 40 do printf(`%d, `, 2*fibonacci(n)^2*fibonacci(n+1)^2) od:
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PROG
| (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); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 08 2009]
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CROSSREFS
| Cf. A001654, A006253, A004003, A006125.
Sequence in context: A062733 A180687 A026739 * A194499 A009478 A038057
Adjacent sequences: A060632 A060633 A060634 * A060636 A060637 A060638
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KEYWORD
| nonn
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AUTHOR
| Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 16 2001
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 16 2001
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