cp's OEIS Frontend

A378724 The number of root fires on a rooted undirected infinite ternary tree with a self-loop at the root, when the chip-firing process starts with 3n chips at the root.

%I A378724 #23 Apr 04 2025 06:29:02
%S A378724 0,1,2,3,5,6,7,9,10,11,13,14,15,18,19,20,22,23,24,26,27,28,31,32,33,
%T A378724 35,36,37,39,40,41,44,45,46,48,49,50,52,53,54,58,59,60,62,63,64,66,67,
%U A378724 68,71,72,73,75,76,77,79,80,81,84,85,86,88,89,90,92,93,94,98,99,100,102,103,104,106,107
%N A378724 The number of root fires on a rooted undirected infinite ternary tree with a self-loop at the root, when the chip-firing process starts with 3n chips at the root.
%C A378724 Each vertex of this tree has degree 4. If a vertex has at least 4 chips, the vertex fires and one chip is sent to each neighbor. The root sends 1 chip to its three children and one chip to itself.
%C A378724 The order of the firings doesn't affect the number of firings.
%C A378724 The corresponding sequence for a binary tree is A376116.
%C A378724 The corresponding sequence for a 4-ary tree is A378726.
%H A378724 Yifan Xie, <a href="/A378724/b378724.txt">Table of n, a(n) for n = 1..10000</a>
%H A378724 Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See p. 10.
%H A378724 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chip-firing_game">Chip-firing game</a>.
%e A378724 Suppose we start with 12 chips at the root. Then the root will fire 3 times, 12 chips total, three of which return to the root. The stable configuration will have 3 chips at the root and at every child of the root. Thus, a(4) = 3.
%e A378724 Suppose we start with 15 chips at the root. Then the root fires 3 times, 12 chips total, sending away 9 chips. Then the root can fire again, sending away 3 chips and keeping 3 chips. Now, each child of the root has four chips and can fire, returning to the root three more chips. Thus, the root can fire one more time. The stable configuration will have 3 chips at the root and 1 chip at each child and grandchild. Thus, a(5) = 5.
%o A378724 (Python)
%o A378724 from math import floor,log
%o A378724 def to_base(number, base): # Converts number to a base
%o A378724    digits = []
%o A378724    while number:
%o A378724       digits.append(number % base)
%o A378724       number //= base
%o A378724    return list(digits)
%o A378724 def c(m,k,convert): # Calculates the c function
%o A378724    try:
%o A378724       num = to_base(convert,k)[m]
%o A378724    except:
%o A378724       num = 0
%o A378724    return num+1
%o A378724 def F(N,k): # Calculated the F function
%o A378724    n = floor(log(N*(k-1)+1)/log(k))
%o A378724    convert = N - int((k**n-1)/(k-1))
%o A378724    ans = 0
%o A378724    for j in range(1,n):
%o A378724       ans += (k**j-1)*c(j,k,convert)
%o A378724    return int(ans/(k-1))
%o A378724 seq = []
%o A378724 for i in range(1,3*100+1,3): # Change this number to get more terms in the sequence
%o A378724    seq.append(F(i+1,3))
%o A378724 print(', '.join(map(str,seq)),end='\n\n')
%Y A378724 Cf. A376116, A378725, A378726, A378727, A378728.
%K A378724 nonn
%O A378724 1,3
%A A378724 _Tanya Khovanova_ and the MIT PRIMES STEP senior group, Dec 05 2024