A352226 Consider a 2D sandpile model where each site with 2 or more grains, say at location (x, y), topples and transfers one grain of sand to the sites at locations (x+1, y) and (x, y+1). Let S(n) be the configuration after stabilization of a configuration with n grains at the origin. a(n) = Max_{ (x,y) in S(n) } (x+y).

0, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23

Offset

1,4

Comments

Sites containing 0 or 1 grain are stable. S(n) contains n elements.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Representation of the configuration for n = 100000

Wikipedia, Sandpile models on directed graphs

Example

For n = 15:
- S(15) corresponds to the following configuration:
    4|  X X X
    3|X   X   X
    2|X     X X
    1|X       X
    0|X X X X
     +---------
      0 1 2 3 4
- x+y is maximized for (x,y) = (4,3) and (3,4),
- so a(15) = 3+4 = 7.

Prog

(PARI) a(n) = { my (s=[n]); for (k=-1, oo, if (vecmax(s)==0, return (k), s \= 2; s = concat(0, s) + concat(s, 0); if (#s>2 && s[1]==0, s = s[2..#s-1]))) }

Crossrefs

Cf. A180230, A349990, A350188, A351783.

Sequence in context: A190268 A111756 A130821 * A129756 A339971 A156724

Adjacent sequences: A352223 A352224 A352225 * A352227 A352228 A352229

Keyword

nonn

Author

Rémy Sigrist, Mar 08 2022

Status

approved