A361457 Numbers k such that the first player has a winning strategy in the game described in the Comments.

3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 40, 42

Offset

1,1

Comments

The game starts with the numbers 1 to k-1. The two players alternate moves. Each move consists of removing two or three numbers that sum to k. The first player with no possible move loses.

Links

Table of n, a(n) for n=1..30.

D. S. et al, Colour 2 or 3 numbers that total 15, Mathematics StackExchange, Mar. 2023.

Example

5 is not a term of the sequence because the first player does not have a winning strategy: the game starts with {1, 2, 3, 4}, the first player must remove either {1, 4} or {2, 3}, the second player removes the remaining two numbers, and the first player has no possible move.
a(3) = 6 is a term because the first player can remove {1, 2, 3}, and the second player has no possible move.

Maple

Filter:= proc(n) local G;
  G:= proc(S) option remember; local nS, i, j, x, k;
    nS:= nops(S);
    for i from 1 to nS-1 do
      for j from i+1 to nS do
        x:= n-S[i]-S[j];
        if x = 0 then if not G(S minus {S[i], S[j]}) then return true fi; break
        elif x > 0 then if member(x, S, 'k') and k > j then if not G(S minus {S[i], S[j], x}) then return true fi fi
        else break
        fi
    od od;
    false
  end proc;
  G({$1..n-1})
end proc:
select(Filter, [$1..40]);

Mathematica

Filter[n_] := Module[{G}, G[S_] := G[S] = Module[{nS, i, j, x, k}, nS = Length[S]; For[i = 1, i <= nS-1, i++, For[j = i+1, j <= nS, j++, x = n-S[[i]]-S[[j]]; If[x == 0, If[!G[S ~Complement~ {S[[i]], S[[j]]}], Return@True]; Break[], If[x > 0, k = FirstPosition[S, x]; If[k != {} && k[[1]] > j, If[!G[S ~Complement~ {S[[i]], S[[j]], x}], Return@True]], Break[]]]]]; False]; G[Range[n-1]]];
Select[Range[30], Filter] (* Jean-François Alcover, May 12 2023, after Robert Israel *)

Prog

(Python)
from functools import lru_cache
from itertools import chain, combinations as C
def t(s): return tuple(sorted(s))
def ok(n):
    def m(S, n):
        yield from (c for c in chain(C(S, 2), C(S, 3)) if sum(c) == n)
    @lru_cache(maxsize=None)
    def winning(S, n):
        return not all(winning(t(set(S)-set(P)), n) for P in m(S, n))
    return winning(tuple(range(1, n)), n)
print([k for k in range(1, 26) if ok(k)]) # Michael S. Branicky, May 19 2023

Crossrefs

Sequence in context: A275481 A234349 A042954 * A247987 A062975 A276218

Adjacent sequences: A361454 A361455 A361456 * A361458 A361459 A361460

Keyword

nonn,more

Author

Robert Israel, Mar 12 2023

Extensions

a(30) from Michael S. Branicky, May 22 2023

Status

approved