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If your opponent has just removed 1 or 2 stones from the pile leaving you with 8, then you lose. Any fewer stones after your opponent has taken 2 will be a win for you.
4, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640, 1659977, 2291229, 3162532, 4365172
If your opponent has just removed 1 or 2 stones from the pile leaving you with 8, then you lose. Any fewer stones after your opponent has taken 2 will be a win for you.
(Python)
MAXTERM=10**9
cache, oldk = [MAXTERM], 1
for nleft in range(1, MAXTERM+1):
for k in range(1, nleft+1):
if k<cache[nleft-k]:
mk=(k+2)//3
break
cache.append(mk)
if mk>oldk:
print(nleft)
oldk=mk
# Bert Dobbelaere, Apr 07 2024
nonn,more
nonn
More terms from Bert Dobbelaere, Apr 07 2024
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R. K. Guy, Fair Game: How to play impartial combinatorial games, COMAP's Mathematical Exploration Series, 1989; see p. 22.
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In Fibonacci Nim , the first player takes any number of stones (except all) and then each player takes no more than twice the amount number taken by in the previous move. This sequence concerns the game where 2 is replaced by 3.
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