Given 2 × 2 tile and seven dominoes (2 × 1 tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlap of any two tiles) 2 × 7 rectangle using some of these tiles.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. an = 2an – 2 + an – 1.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. a1 = 1, a2 = 3.
Hint 3: Proceed with the final algebraic steps to solve the system. a3 solve for the final valuea1 + a2 solve for the final value(1) + 3 solve for the final value.
Step 1: an = 2an – 2 + an – 1
Step 2: a1 = 1, a2 = 3
Step 3: a3 = 2a1 + a2 = 2(1) + 3 = 5
Step 4: a4 = 2a2 + a3 = 2(3) + 5 = 11
Step 5: a5 = 2a3 + a4 = 2(5) + 11 = 21
Step 6: a6 = 2(11) + 21 = 43
Step 7: a7 = 2(21) + 43 = 42 + 43 = 85
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