cp's OEIS Frontend

A378727 The total number of fires in a rooted undirected infinite 4-ary tree with a self-loop at the root, when the chip-firing process starts with (4^n-1)/3 chips at the root.

%I A378727 #18 Apr 02 2025 20:14:05
%S A378727 0,1,10,67,380,1973,9710,46119,213600,970905,4349650,19262731,
%T A378727 84507460,367855997,1590728630,6840133103,29269406760,124713124449,
%U A378727 529394487450,2239745908435,9447655468300,39745309211461,166799986198910,698474942207927,2918999758480880,12176398992520233,50707195804467810
%N A378727 The total number of fires in a rooted undirected infinite 4-ary tree with a self-loop at the root, when the chip-firing process starts with (4^n-1)/3 chips at the root.
%C A378727 Each vertex of this tree has degree 5. If a vertex has at least 5 chips, the vertex fires, and one chip is sent to each neighbor. The root sends 1 chip to each of its four children and one chip to itself.
%C A378727 The order of the firings doesn't affect the number of firings.
%C A378727 This number of chips is interesting because the stable configuration has 1 chip for every vertex in the top n layers.
%C A378727 a(n) is partial sums of A014916.
%C A378727 For binary trees, the corresponding sequence is A045618.
%C A378727 For ternary trees, the corresponding sequence is A212337.
%C A378727 For 5-ary trees, the corresponding sequence is A378728.
%C A378727 a(2k-1) is divisible by 10.
%H A378727 Yifan Xie, <a href="/A378727/b378727.txt">Table of n, a(n) for n = 1..2000</a>
%H A378727 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. 15.
%H A378727 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chip-firing_game">Chip-firing game</a>.
%H A378727 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-33,40,-16).
%F A378727 a(n) = ((3*n - 5)*4^n + 3*n + 5)/27.
%t A378727 Table[((3 n - 5) 4^n + 3 n + 5)/27, {n, 30}]
%Y A378727 Cf. A045618, A014916, A212337, A378728.
%K A378727 nonn,easy
%O A378727 1,3
%A A378727 _Tanya Khovanova_ and the MIT PRIMES STEP senior group, Dec 05 2024