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A350188 Consider a 2D sandpile model where each site with 3 or more grains, say at location (x, y), topples and transfers one grain of sand to the sites at locations (x+1, y-1), (x+1, y) and (x+1, y+1); a(n) gives the number of nonempty sites after stabilization of a configuration starting with n grains at the origin. 2
0, 1, 1, 3, 4, 4, 3, 4, 4, 7, 8, 8, 10, 11, 11, 10, 11, 11, 14, 15, 15, 17, 18, 18, 17, 18, 18, 21, 22, 22, 24, 25, 25, 24, 25, 25, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40, 40, 39, 40, 40, 39, 40, 40, 42, 43, 43, 42, 43, 43, 45, 46, 46, 48, 49 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Sites containing 0, 1 or 2 grains are stable.
After stabilization, there are:
- 2*a(n) - n sites with one grain,
- n - a(n) sites with two grains.
LINKS
Rémy Sigrist, Representation of the configuration for n = 100000 (blue pixels correspond to sites with one grain, red pixels to sites with two grains)
FORMULA
a(3*n) + 1 = a(3*n + 1) = a(3*n + 2).
EXAMPLE
For n = 10 :
- the model evolves (for example) as follows:
1 1
3 . 2 . 2 1
10 -> 1 3 -> 1 . 3 -> 1 . . 1
3 . 2 . 2 1
1 1
- there are 8 nonempty sites in the stabilized configuration,
- so a(10) = 8.
PROG
(PARI) a(n) = { my (s=[n], v=0); for (k=-1, oo, if (vecmax(s)==0, return (v), v += sum(k=1, #s, s[k]%3>0); s \= 3; s = concat([ s , [0], [0]]) + concat([[0], s , [0]]) + concat([[0], [0], s ]); while (#s>2 && s[1]==0, s = s[2..#s-1]))) }
CROSSREFS
Sequence in context: A306477 A276865 A253718 * A295570 A035004 A222290
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Mar 09 2022
STATUS
approved

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Last modified May 30 17:31 EDT 2024. Contains 372971 sequences. (Running on oeis4.)