[N. J. Sloane, Feb 13 2020: I asked Alex Fink if this sequence was discussed in any of his papers, and he replied as follows.]

I remember the outline of what we were trying to do when inventing this.  I don't think we got any striking results, and I don't believe it made it to a paper.

Subtraction games, in the usual sense in combinatorial game theory,
can be reformulated as string-cutting games as in the description of
A119346 with the same subtraction set, now interpreted as the lengths
of bits of string that can be cut off in a move.  If the subtraction
set is a set of integers then the fractional part of the length of the
starting string doesn't matter, i.e. the Grundy function is constant
on intervals [k, k+1) where  k  is an integer.  The scale-invariance
of subtraction games extends to these string games: if G_S is the
Grundy function of the game with subtraction set S, then G_(aS)(ax) =
G_S(x) for any positive real  a.

As I remember, at least some known general results for Grundy
functions of subtraction games -- there weren't many -- extended
nicely to arbitrary positive real subtraction sets S.  This includes
the description when |S| <= 2, which includes the case of this
sequence.  The string game with S = {1, sqrt(2)} has Grundy function
of period 1 + sqrt(2): it takes value 0 on [0, 1), 1 on [1, 2), 2 on
[2, 1+sqrt(2) ), and then repeats.


But the reason we were doing this was an attempt to get a complete
description for the case |S| = 3.  Say S = {a, b, c}.  Given a natural
 n,  suppose we're interested only in the first  n  intervals on which
the Grundy function is piecewise constant.  Then there is a cone
complex in (a, b, c)-space so that, in each cone, the lengths of these
 n  intervals are fixed linear functions of a, b, c, and the list of
successive Grundy values taken on the intervals is constant.  As  n
increases, this cone complex gets refined (though some individual
cones, where G has become periodic, stop being refined).  Perhaps we
could describe its limit?

Not surprisingly, this proved hard.  In retrospect, the combinatorics
of the situation (not that I remember it at all clearly!) was roughly
reminiscent of that of multidimensional continued fractions, with the
usual added Grundy sort of complexity that only in some regions of the
complex do new small linear combinations of  a, b, c  actually hit the
current "candidate" period of the Grundy function in the right way to
disrupt it.

The simplest description of the sequence , without invoking games, is
that A119346(n) = floor(n mod (1 + sqrt 2)).

The relation with the Beatty sequence A003151 is that, if  m = ceil(k
* (1 + sqrt 2)) = 1 + A003151(k) for some positive integer  k, then 0
< m mod (1 + sqrt 2) < 1.  That is, A119346(n) = 0 if  n  is 1 + a
member of A003151.  For any positive integer  n  not of this form, we
have (n - 1) mod (1 + sqrt 2) = (n mod (1 + sqrt 2)) - 1.  Those two
facts are what the definition of A119346 are attempting to convey.