MATL, 12 bytes
tnEXyPY+c!Us
Input is a string. Try it online!
###Explanation
Explanation
Consider input '123' as an example.
The code duplicates the input (t) and builds an identity matrix (Xy) of size twice the input length (nE):
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
then flips it upside down (P):
0 0 0 0 0 1
0 0 0 0 1 0
0 0 0 1 0 0
0 0 1 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
The input string, interpreted as ASCII codes of the digits, is equivalent to the numeric row vector
49 50 51
Full-size two-dimensional convolution (Y+) of the above vector and matrix gives
0 0 0 0 0 49 50 51
0 0 0 0 49 50 51 0
0 0 0 49 50 51 0 0
0 0 49 50 51 0 0 0
0 49 50 51 0 0 0 0
49 50 51 0 0 0 0 0
Interpreting those numbers back as ASCII codes (c) gives the following char matrix, with char 0 represented as space:
123
123
123
123
123
123
Transposition (!) transforms this into
1
12
123
123
123
123
23
3
Interpreting each row as a number (U) gives the numeric column vector
1
12
123
123
123
123
23
3
and summing it (s) gives the final result, 531.
