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%I #23 Sep 25 2021 07:25:36
%S 193,399,783,1601,3283,6947,14897,32607,72175,161649,364611,827555,
%T 1885729,4310639,9874319,22654881,52032883,119601123,275058321,
%U 632823743,1456319215,3352072913,7716633443,17765737443,40904125825,94182711375
%N Number of ways to tile an n X n square with 1 X 1 squares and (n-2) X 2 vertical or horizontal rectangles.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-7,1,3).
%F a(n) = 2*A006130(n) + 12*F(n + 1) + 16*F(n - 1) - 31 for F(n) = A000045(n) the Fibonacci sequence.
%F a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + a(n-4) + 3*a(n-5).
%e Here are two of the 193 possible tilings for a 5 X 5 square (using 1 X 1 squares and 3 X 2 rectangles):
%e ._________ ._________
%e |_| |_| |_|_| |
%e |_|_ _ _|_| | |_ _ _|
%e | |_| | | |_| |
%e | |_| | |___|_| |
%e |___|_|___| |_|_|_|___|
%Y Cf. A000045, A006130.
%Y Cf. A335560 which is the same problem but with 1 X 1 squares and (n-1) X 1 rectangles, and A337024 which uses 1 X 1 squares and 2 X 2 squares.
%K nonn,easy
%O 5,1
%A _Greg Dresden_ and _Osondu Ugochukwu_, Sep 09 2021