A378725 a(n) = A378724(n+1) - A378724(n).

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3

Offset: 1

Author

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

Keywords

• nonn

Comments

a(n) is 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 3(n+1) chips at the root minus the number of root fires in the same tree, when a chip-firing process starts with 3n chips at the root.

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

Examples

			Suppose we start with 12 chips at the root. Then the root will fire 3 times, 12 chips in total, 3 of which return to the root. The stable configuration will have 3 chips at the root and at every child of the root. Thus, the root fires 3 times in total.
Suppose we start with 15 chips at the root. Then the root will fire 3 times, 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 they can also fire. Firing them returns three chips to the root. 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, the root fires 5 times. It follows that a(4) = 5-3 = 2.
		

References

• The difference sequence for binary trees is 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. 11.
• Wikipedia, Chip-firing game.

Crossrefs

Cf. A091090, A378724, A378726, A378727, A378728.

Programs

Mathematica

c[n_] := c[n] = Which[n == 1, 1, Mod[n, 3] != 1, 1, True, c[(n - 1)/3] + 1]; Array[c, 103, 1]