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A347636 Number of ways to tile an n X n square with 1 X 1 squares and (n-2) X 2 vertical or horizontal rectangles.

%I A347636 #23 Sep 25 2021 07:25:36
%S A347636 193,399,783,1601,3283,6947,14897,32607,72175,161649,364611,827555,
%T A347636 1885729,4310639,9874319,22654881,52032883,119601123,275058321,
%U A347636 632823743,1456319215,3352072913,7716633443,17765737443,40904125825,94182711375
%N A347636 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 A347636 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-7,1,3).
%F A347636 a(n) = 2*A006130(n) + 12*F(n + 1) + 16*F(n - 1) - 31 for F(n) = A000045(n) the Fibonacci sequence.
%F A347636 a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + a(n-4) + 3*a(n-5).
%e A347636 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 A347636 ._________   ._________
%e A347636 |_|     |_|  |_|_|     |
%e A347636 |_|_ _ _|_|  |   |_ _ _|
%e A347636 |   |_|   |  |   |_|   |
%e A347636 |   |_|   |  |___|_|   |
%e A347636 |___|_|___|  |_|_|_|___|
%Y A347636 Cf. A000045, A006130.
%Y A347636 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 A347636 nonn,easy
%O A347636 5,1
%A A347636 _Greg Dresden_ and _Osondu Ugochukwu_, Sep 09 2021