cp's OEIS Frontend

A378728 The total number of fires in a rooted undirected infinite 5-ary tree with a self-loop at the root, when the chip-firing process starts with (5^n-1)/4 chips at the root.

%I A378728 #34 Apr 04 2025 23:42:57
%S A378728 0,1,12,98,684,4395,26856,158692,915528,5187989,28991700,160217286,
%T A378728 877380372,4768371583,25749206544,138282775880,739097595216,
%U A378728 3933906555177,20861625671388,110268592834474,581145286560060,3054738044738771,16018748283386232,83819031715393068
%N A378728 The total number of fires in a rooted undirected infinite 5-ary tree with a self-loop at the root, when the chip-firing process starts with (5^n-1)/4 chips at the root.
%C A378728 Each vertex of this tree has degree 6. If a vertex has at least 6 chips, the vertex fires, and one chip is sent to each neighbor. The root sends 1 chip to each of its five children and one chip to itself.
%C A378728 The order of the firings doesn't affect the number of firings.
%C A378728 This number of chips is interesting because the stable configuration has 1 chip for every vertex in the top n layers.
%C A378728 a(n) is partial sums of A014917.
%C A378728 For binary trees, the corresponding sequence is A045618.
%C A378728 For ternary trees, the corresponding sequence is A212337.
%C A378728 For 4-ary trees, the corresponding sequence is A378727.
%C A378728 a(2k-1) is divisible by 12.
%H A378728 Yifan Xie, <a href="/A378728/b378728.txt">Table of n, a(n) for n = 1..1000</a>
%H A378728 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. 16.
%H A378728 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chip-firing_game">Chip-firing game</a>.
%H A378728 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-46,60,-25).
%F A378728 a(n) = ((2*n - 3)*5^n + 2*n + 3)/32.
%F A378728 G.f.: x^2/(1-6*x+5*x^2)^2. - _Jinyuan Wang_, Jan 24 2025
%t A378728 Table[((2 n - 3) 5^n + 2 n + 3)/32, {n, 30}]
%Y A378728 Cf. A014917, A045618, A212337, A378727.
%K A378728 nonn,easy
%O A378728 1,3
%A A378728 _Tanya Khovanova_ and the MIT PRIMES STEP senior group, Dec 05 2024