Number of Straight-Chain Alk*nes of given length

MATL, 10 bytes

7K5vBiY^1)

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Explanation

This uses the characterization found in OEIS

a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 0, 0; 1, 0, 1]

7    % Push 7
K    % Push 4
5    % Push 5
v    % Concatenate all numbers into a column vector: [7; 4; 5]
B    % Convert to binary: gives 3×3 matrix [1, 1, 1; 1, 0, 0; 1, 0, 1]
i    % Input n
Y^   % Matrix power
1)   % Take the first element of the resulting matrix, i.e. its upper-left corner.
     % Implicitly display

Oasis, 9 7 bytes

xcd-+3V

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Explanation

This uses the recurrence relationship in OEIS:

a(n) = 2*a(n-1) + a(n-2) - a(n-3)

x    Multiply a(n-1) by 2: gives 2*a(n-1)
c    Push a(n-2)
d    Push a(n-3)
-    Subtract: gives a(n-2) - a(n-3)
+    Add: gives 2*a(n-1) + a(n-2) - a(n-3)
3    Push 3: initial value for a(n-1)
V    Push 1, 1: initial values for a(n-2), a(n-3)

Oasis, 9 8 bytes

Saved a byte thanks to Adnan

xc+d-63T

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Explanation

a(0) = 0
a(1) = 1
a(2) = 3
a(3) = 6

a(n) = xc+d-

x         # calculate 2*a(n-1)
 c        # calculate a(n-2)
  +       # add: 2*a(n-1) + a(n-2)
   d      # calculate a(n-3)
    -     # subtract: 2*a(n-1) + a(n-2) - a(n-3)