login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A180230 a(n) is the minimal number of additions needed to grow to radius n, in the two-dimensional abelian sandpile growth model with h=2. 1
2, 6, 10, 22, 26, 50, 66, 78, 122, 142, 154, 194, 254, 270, 342, 386, 418, 490, 518, 578, 654, 698, 766, 914, 942, 1074, 1150, 1178, 1310, 1366, 1410, 1570, 1646, 1794, 1894 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The abelian sandpile growth model starts with height h on every site of the square grid.

An addition increases the height of the origin by 1. After each addition, the model is stabilized by toppling unstable sites.

A site is unstable if its height is at least 4; in a toppling, its height decreases by 4 and the height of its neighbors increases by 1.

If h=2, then for any number of additions, the set of sites that toppled at least once is a square. This was proved in Fey-Redig-2008.

For all n, a(n) <= (2n+3)^2. In Fey-Levine-Peres-2010, it was proved that for n large enough, a(n) >= Pi/4 n^2.

REFERENCES

Anne Fey, Lionel Levine and Yuval Peres, Growth rates and explosions in sandpiles, Journal of Statistical Physics 138 (2010), 143-159.

Anne Fey and Frank Redig, Limiting shapes for deterministic centrally seeded growth models , Journal of Statistical Physics 130 (2008), 579-597.

LINKS

Table of n, a(n) for n=0..34.

Anne Fey, MATLAB program

EXAMPLE

After 2 additions, the origin is unstable and topples once. Then every site is stable. Therefore a(0)=2.

After 4 more additions, the origin topples again. Then more sites become unstable, so that the set of sites that toppled at least once becomes the square with radius 1. Therefore a(1) = 6.

CROSSREFS

Cf. A056219

Sequence in context: A272952 A034450 A112861 * A186296 A140775 A077064

Adjacent sequences:  A180227 A180228 A180229 * A180231 A180232 A180233

KEYWORD

nonn

AUTHOR

Anne Fey (a.c.fey-denboer(AT)tudelft.nl), Aug 17 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 20 08:12 EST 2017. Contains 294962 sequences.