A054736 - OEIS

A054736

Smallest losing position after your opponent has taken k stones in a variation of "Fibonacci Nim".

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

In Fibonacci Nim, the first player takes any number of stones (except all) and then each player takes no more than twice the number taken in the previous move. This sequence concerns the game where 2 is replaced by 3.

REFERENCES

R. K. Guy, Fair Game: How to play impartial combinatorial games, COMAP's Mathematical Exploration Series, 1989; see p. 22.

LINKS

Table of n, a(n) for n=1..43.

EXAMPLE

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.

PROG

(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)