Searching for a maximum-sum subsequence

Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is $16 (= 8 + 7 + 1)$.

-2	5	3	2
9	-6	5	1
3	2	7	3
-1	8	-4	8

First, generate four million pseudo-random numbers using a specific form of what is known as a "Lagged Fibonacci Generator":

For $1 ≤ k ≤ 55, s_k = [100003 − 200003k + 300007k^3]$ (modulo $1000000$) − $500000$. For $56 ≤ k ≤ 4000000$, $s_k = [s_{k−24} + s_{k−55} + 1000000$] (modulo $1000000$) − $500000$.

Thus, $s_{10} = −393027$ and $s_{100} = 86613$.

The terms of s are then arranged in a $2000×2000$ table, using the first $2000$ numbers to fill the first row (sequentially), the next $2000$ numbers to fill the second row, and so on.

Finally, find the greatest sum of (any number of) adjacent entries in any direction (horizontal, vertical, diagonal or anti-diagonal).