TITLE: What is the maximum amount of "shore" tiles in a rectangular grid that has only "water" and "land" tiles?
QUESTION [6 upvotes]: I'm not a mathematician or anything of the like, so please forgive me if I'm not using the correct terminology for something.
The problem is as follows:

Given a rectangular grid that has $m*n$ tiles, where $m$ is the amount of rows and $n$ is the amount of columns in the grid, and knowing that:

$1 \leq m \leq 100$
$1 \leq n \leq 100$
Every tile is classified as either water or land
The grid can have any amount of water tiles ranging from $0$ to $m*n$
A shore tile is defined as any land tile that has at least 1 of its 4 (or 3 on the edges, or 2 on the corners) adjacent tiles as a water tile

What is the maximum amount of shore tiles a grid can have (I'll call it $R$)?

I started working on this by first trying to solve for very small grids and go up from there, so, for a 1x1 grid, $R = 0$, since there can't be any adjacent tile to that single tile.
W: water tile
L: land tile
$s$: amount of shore tiles
Then, for 1x2 grids:




WW
WL
LL




$s=0$
$s=1$
$s=0$




So, for 1x2 (and 2x1) grids, $R = 1$
For 1x3 grids:




WWW
WWL




$s=0$
$s=1$




(ignoring some other possibilities with 1 L, because I'm interested in the maximum value, and $s$ can't be higher than the amount of L in the grid)




WLL
LWL
LLL




$s=1$
$s=2$
$s=0$




So, for 1x3 (and 3x1) grids, $R = 2$
(...)
For 2x3 grids:




WWWWWW
WWWWWL
WWWWLL
WWWLLL
WWLLLL
LWLLWL
WLLLLL
LWLLLL
LLLLLL




$s$=0
$s$=1
$s$=2
$s$=3
$s$=3
$s$=4
$s$=2
$s$=3
$s$=0




(with only 2 rows or columns, $s \leq 3*\text{amount of W's}$, and reaching the maximum has been confirmed to be possible here)
So, for 2x3 (and 3x2) grids, $R = 4$

|m \ n|1|2|3|4|5|6|
|-----|-|-|-|-|-|-|
|  1  |0|1|2|2|3|4|
|  2  |1|2|4|?|?|?|
|  3  |2|4|?|?|?|?|
|  4  |2|?|?|?|?|?|
|  5  |3|?|?|?|?|?|
|  6  |4|?|?|?|?|?|

So, up until now
$$R = m*n - 1 - \lfloor m*n/4 \rfloor$$
But this is likely not a general formula (right?).
Is there a known general solution for mxn grids? And if so, is there an easier way for getting to it than what I'm doing?
Edit: reduced amount of examples, tried to fix the table at the end.
Edit 2: since it seems like the preview and the actual table don't match, I'll leave the table preformatted for now.

REPLY [2 votes]: Let us present on examples a graphical understanding of how sub-optimal solutions can be constructed. This looks to be in the same 'spirit' as the answer given by @Hagen von Eitzen:

Fig. 1: Case $m=10, n=7$. Yellow: sand, Blue: water (of course...). The land, if any, is brown. Not yet optimal (see Fig. 2)...

Fig. 2: Looks good... but is still far from optimality.
With width-2 sand strips with a 45° slope, separated by width-1 water strips as displayed on Fig. 2, we have 46 yellow "tiles" for a total of $70$, with an approximate proportion $\approx 0.66\%$.
One can in fact get better arrangements as remarked by @Tanner Swett by placing the water tiles in a "quincunx" way (think to "knight's moves") like the one of figure 3, necessitating small adjustments at the boundary. This possibility comes from the paving of an infinite plane by greek crosses (the "plus" shapes of the answer by HvE, which are the "influence zone" of a water tile) as shown on fig. 4 where the places where adjustments have to be made are indicated by red crosses.

Fig. 3: Case $m=13, n=11$. Here, $106$ yellow tiles among $143$, otherwise said $\approx 74\%$ of the total number of tiles. Most horizontal strips display a "trend" of 4 yellow tiles among $5$; therefore, the density of yellow tiles will asymptotically be $80\%$ of the total number of tiles as explained in the solution by HvE.

Fig. Tiling of a rectangle (red boundary) with greek crosses.