A390129 The number of layers of vertices that ever have chips in a certain chip-firing game that starts with 2^n chips and is described in the comments.

1, 2, 4, 6, 10, 16, 24, 38, 60, 92, 144, 226, 362, 570, 906, 1430, 2272, 3600, 5710, 9072, 14390, 22838, 36254, 57562, 91354, 145028, 230102, 365296, 579822, 920430, 1460994, 2319002, 3681214, 5843546, 9276188, 14724728, 23373826, 37103118

Offset

0,2

Comments

Consider the following chip-firing game on the integer points of the first quadrant (i.e., set of vertices (x, y) where x and y are nonnegative integers): start with 2^n chips at the origin (i.e., point (0, 0)). If a vertex (x, y) has at least 2 chips, it fires by sending one chip to (x+1, y) and one chip to (x, y+1). The vertex (x, y) loses two chips during each firing. The total number of firings in the game is finite. For each nonnegative integer i, we define layer i to be the set of all vertices that have Manhattan distance i from the origin. The term a(n) is the number of layers that ever have chips.

Beginning with the term at index n = 1, this sequence of integers is even.

Below, we have a table for the maximum number of chips at each vertex in the firing process for n=4:

16

8 8

4 8 4

2 6 6 2

1 4 6 4 1

2 5 5 2

1 3 4 3 1

1 3 3 1

1 2 1

1 1

References

Caroline Klivans, The Mathematics of Chip-Firing, CRC Press, 2019, Chapter 6.5.

Links

Table of n, a(n) for n=0..37.

Ryota Inagaki, Tanya Khovanova, and Austin Luo, Chip-firing on the Lattice of Nonnegative Integer Points, arXiv:2601.09125 [math.CO], 2026. See p. 12.

Wikipedia, Chip-firing game.

Prog

(Python)
def generate_and_count_rows(n):
    l1 = [2**n]
    stab = (max(l1) < 2)
    layer_count = 1
    while not stab:
        l2 = [l1[0]//2]
        for i in range(1, len(l1)):
            l2.append(l1[i]//2 + l1[i-1]//2)
        l2.append(l1[len(l1)-1]//2)
        l1 = l2
        if (max(l1) < 2):
            stab = True
        if l1[0] == 0:
            l1 = l1[1:]
        if l1[-1] == 0:
            l1 = l1[:len(l1)-1]
        layer_count+= 1
    return layer_count
for i in range(28):
    print(generate_and_count_rows(i))

Crossrefs

Cf. A390355, A389565, A259013, A376131, A378726.

Sequence in context: A211971 A305498 A028445 * A006305 A067247 A017985

Adjacent sequences: A390126 A390127 A390128 * A390130 A390131 A390132

Keyword

nonn,more

Author

Ryota Inagaki, Tanya Khovanova, and Austin Luo, Nov 28 2025

Extensions

a(28)-a(37) from Sean A. Irvine, Dec 10 2025

Status

approved