Restricted Nim and fractal sequences
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Maximum Nim & Minimum Nim

One considers a 1-graph without loops whose set of vertices is X.
A function of Grundy, when it exists, assigns to every vertex x of the graph a nonnegative integer g(x) which is the smallest not in g(Y) where Y is the set of the successors of X (P. Mr. Grundy 1939).
The function g(x) is often noted mex(x) (MEX: Minimal EXcludant)

The game of Nim where the two players remove in turn, 1, 2 or three stones of the heap. The winner is that which withdraws the last stone.
The associated graph have the set X = {0, 1, 2, ..., 200} of vertices and each vertex x have x-1, x-2 et x-3 for successors (when they exists).
g(0) = 0 because 0 have no successor.
g(1) = 1, g(2) = 2, g(3)=3,
g(4) = 0 because g(1)=1, g(2)=2, g(3)=3 and therefore 0 is the smallest non-negative integer not in {1, 2, 3}.
g(n) = n%4 is the remainder of the Euclidean division of N by 4. We obtain the sequence ID Number: A010873 of N.J.A. Sloane

Not always a graph admit a function of Grundy.
A symmetrical graph admits a function of Grundy. Idem for a transitive graph and a graph without circuit odd length. A graph without circuit admits a unique function of Grundy G and for any vertex x, g(x) is lower or equal to the length of the longest path starting at x.
The set N of vertices x such as g(x) = 0 is a kernel of the graph. Indeed, it is easy to show
x in N does not have a successor y in N.
if X is not in N, then it has a successor in N.

Divide-and-conquer (important for the games of Nim with several heaps): if a graph G is a Cartesian sum of several G_i graphs admitting a function of Grundy g_i, then G admits for function of Grundy the digital sum (the XOR noted ~) G = g_1 ~ g_2 ~ g_3..

In the example describes higher, the withdrawal carried out by a player lies between 1 and 3. One will generalize this play by defining a minimum min(n) and a maximum max(n) stones removed depending only on number N of stones of the heap. According to the type of game, remove between min(n) and N stones for minimum Nim and 1 to max(n) stones for maximum Nim. Note: The game of Fibonacci-Nim is not included in this category because the interval depends on the preceding play and not on number N of stones.

Some properties are shown in the references. One will be able to seek to discover them on the examples suggested (to click on the button [ Examples ]).
One can also make new examples.
The calculated sequences are the g(i) for i varying from 0 to n.

Maximum n/2
Minimum n/2
Maximum periodic

You can modify below the values of the two tables this.min[i] or this.max[i].
The code is write in Javascript and you can use the mathematical functions of the Javascript language)
Some of these sequences are referred on the site of N.J.A. Sloane. Pure traditional Nim is: click HERE.

Restricted Nim - Grundy

n =	function of Grundy G   lines of terms   difference n- g(n) (g, f, g-f-1) when f = max[i] grows

for(var i=0; i<=n; i++) { this.min[i] = floor(1) this.max[i] = floor(i/2); }