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.

0, 1, 10, 67, 380, 1973, 9710, 46119, 213600, 970905, 4349650, 19262731, 84507460, 367855997, 1590728630, 6840133103, 29269406760, 124713124449, 529394487450, 2239745908435, 9447655468300, 39745309211461, 166799986198910, 698474942207927, 2918999758480880, 12176398992520233, 50707195804467810

Offset: 1

Author

Tanya Khovanova and the MIT PRIMES STEP senior group, Dec 05 2024

Keywords

• nonn
• easy

Comments

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.

The order of the firings doesn't affect the number of firings.

This number of chips is interesting because the stable configuration has 1 chip for every vertex in the top n layers.

a(n) is partial sums of A014916.

For binary trees, the corresponding sequence is A045618.

For ternary trees, the corresponding sequence is A212337.

For 5-ary trees, the corresponding sequence is A378728.

a(2k-1) is divisible by 10.

Links

• Yifan Xie, Table of n, a(n) for n = 1..2000
• 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. 15.
• Wikipedia, Chip-firing game.
• Index entries for linear recurrences with constant coefficients, signature (10,-33,40,-16).

Crossrefs

Cf. A045618, A014916, A212337, A378728.

Programs

Mathematica

Table[((3 n - 5) 4^n + 3 n + 5)/27, {n, 30}]

Formula

a(n) = ((3*n - 5)*4^n + 3*n + 5)/27.