Sum of neighbours
Python, 25 bytes
lambda a:sum((a*3)[1:-1])
To see why this works, rotate the expansion in the OP by 45 degrees:
1 + 2
+ 1 + 2 + 3 2 + 3 + 4 + 5
+ 2 + 3 + 4 = + 1 + 2 + 3 + 4 + 5
+ 3 + 4 + 5 + 1 + 2 + 3 + 4.
+ 4 + 5
Python 2, 28 bytes
lambda a:sum(a)*3-a[0]-a[-1]
Just 3 times the sum and minus one of each end element
MATL, 5 bytes
7BZ+s
Try it online!
Explanation
7B % Push array [1, 1, 1], obtained as 7 in binary
Z+ % Take input implicitly. Convolve with [1, 1, 1], keeping size
s % Sum of resulting array. Display implicitly