OFFSET
1,3
COMMENTS
Adding a loop at the root makes the graph 3-regular: each vertex has degree 3.
The first differences of this sequence give A091090.
LINKS
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 10.
Wikipedia, Chip-firing game.
FORMULA
a(n) = Sum_{j=1..m-1} (2^j-1)(b(j)+1), where m = floor(log_2(2n+1)) and b(m)b(m-1)...b(1)b(0) is the binary representation of 2*n+1.
EXAMPLE
If there are four chips at the root, then the root fires and the process ends in a stable configuration.
If there are eight chips at the root, the root can fire three times, sending 3 chips to each child. After this, each child can fire once. After that the root has 4 chips and can fire again. The root fires a total of 4 times.
MAPLE
a:= n-> (l-> add((2^(i-1)-1)*(l[i]+1), i=2..nops(l)-1))(Bits[Split](2*n+1)):
seq(a(n), n=1..70); # Alois P. Heinz, Sep 12 2024
PROG
(Python)
def a(n):
if n <= 2:
return 0
else:
return (n+1) // 2 - 1 + a((n+1)//2 - 1)
print([a(2*n) for n in range(1, 51)])
(Python)
def A376116(n): return (n<<1)-n.bit_count()-n.bit_length() # Chai Wah Wu, Sep 18 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved