A056219 -id:A056219 - cp's OEIS frontend

A166869 a(n) = n * A056219(n+1).

2, 4, 12, 20, 30, 54, 91, 120, 171, 250, 374, 504, 663, 854, 1170, 1568, 2074, 2628, 3325, 4180, 5313, 6754, 8602, 10656, 13100, 16042, 19683, 24024, 29464, 36000, 43834, 52768, 63228, 75582, 90510, 107856, 128575, 153178, 182208, 215400

Offset: 1

Roger L. Bagula, Oct 22 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A056219, A166870.

max:=50;
R:=PowerSeriesRing(Integers(), max); b:= Coefficients(R!( (&+[x^Binomial(n+1,2)*(&*[x + 1/(1-x^j): j in [1..n]]): n in [1..Floor(Sqrt(9+8*max)/2)]]) ));
[(n-1)*b[n]: n in [2..max-1]]; // G. C. Greubel, Nov 29 2019

N:= 100; b:= seq(coeff(series(add(x^((1/2)*n*(n+1))*mul(x +1/(1-x^k), k=1..n), n = 1..floor((1/2)*sqrt(9+8*N))), x, N+2), x, j), j = 1..N+1); seq(n*b[n+1], n=1..N); # G. C. Greubel, Nov 29 2019

max:= 100; b:= CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[(x +1/(1-x^k)), {k, n}], {n, Floor[Sqrt[9 +8*(max+5)]/2]}], {x, 0, max+5}], x]; Table[n*b[[n + 2]], {n, max}] (* G. C. Greubel, Nov 29 2019 *)

max=50;
def A056219_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sum(x^binomial(n+1,2)*product((x + 1/(1-x^j)) for j in (1..n)) for n in (1..floor(sqrt(9+8*max)/2))) ).list()
b=A056219_list(max);
[(n-1)*b[n] for n in (2..max)] # G. C. Greubel, Nov 29 2019

A166870 a(n) = n(n-1)A056219(n+1).

Original entry on oeis.org

4, 24, 60, 120, 270, 546, 840, 1368, 2250, 3740, 5544, 7956, 11102, 16380, 23520, 33184, 44676, 59850, 79420, 106260, 141834, 189244, 245088, 314400, 401050, 511758, 648648, 824992, 1044000, 1315020, 1635808, 2023296, 2494206, 3077340

Offset: 2

Roger L. Bagula, Oct 22 2009

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A056219, A166869.

max:=50;
R:=PowerSeriesRing(Integers(), max); b:= Coefficients(R!( (&+[x^Binomial(n+1,2)*(&*[x + 1/(1-x^j): j in [1..n]]): n in [1..Floor(Sqrt(9+8*max)/2)]]) ));
[(n-1)*(n-2)*b[n]: n in [3..max-1]]; // G. C. Greubel, Nov 29 2019

N:= 100; b:= seq(coeff(series(add(x^((1/2)*n*(n+1))*mul(x +1/(1-x^k), k=1..n), n = 1..floor((1/2)*sqrt(9+8*N))), x, N+2), x, j), j = 1..N+1); seq(n*(n-1)*b[n+1], n=2..N); # G. C. Greubel, Nov 29 2019

max:= 100; b:= CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[(x +1/(1-x^k)), {k, n}], {n, Floor[Sqrt[9 +8*(max+5)]/2]}], {x, 0, max+5}], x]; Table[n*b[[n + 2]], {n, 2, max}] (* G. C. Greubel, Nov 29 2019 *)

max=50;
def A056219_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sum(x^binomial(n+1,2)*product((x + 1/(1-x^j)) for j in (1..n)) for n in (1..floor(sqrt(9+8*max)/2))) ).list()
b=A056219_list(max);
[(n-1)*(n-2)*b[n] for n in (3..max)] # G. C. Greubel, Nov 29 2019

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.

Original entry on oeis.org

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, 2054, 2130, 2246, 2406, 2466, 2654, 2742, 2894, 3006, 3138, 3318, 3582, 3670, 3826

Offset: 0

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

nonn

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.

			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.

Anne Fey, MATLAB program
Anne Fey, Lionel Levine and Yuval Peres, Growth rates and explosions in sandpiles, arXiv:0901.3805 [math.CO], 2009.
Anne Fey, Lionel Levine and Yuval Peres, Growth Rates and Explosions in Sandpiles, Journal of Statistical Physics, Vol. 138, No. 1-3 (2010), 143-159.
Anne Fey and Frank Redig, Limiting shapes for deterministic centrally seeded growth models, arXiv:math/0702450 [math.PR], 2007.
Anne Fey and Frank Redig, Limiting Shapes for Deterministic Centrally Seeded Growth Models, Journal of Statistical Physics 130 (2008), 579-597.
Rémy Sigrist, C++ program for A180230

Cf. A056219

More terms from Rémy Sigrist, Dec 15 2021

A292726 a(n) is the number of states that can be achieved when starting from n piles each containing one stone, where any number of stones can be transferred between piles that start with the same number of stones.

Original entry on oeis.org

1, 2, 2, 5, 6, 8, 14, 22, 27, 38, 55, 70, 100, 130, 167, 231, 296, 371, 489, 618, 775, 995, 1254, 1549, 1951, 2428, 2980, 3707, 4564, 5549, 6841, 8349, 10085, 12300, 14862, 17894, 21636, 26004, 31082, 37308, 44582, 53024, 63260, 75160, 88931, 105545, 124753, 147034, 173510, 204174

Offset: 1

Peter Kagey, Sep 21 2017

nonn

This sequence is bounded above by A000041.
Conjecture: A000041(n) - a(n) = 0 if and only if n is a power of 2, and
Conjecture: A000041(n) - a(n) = 1 if and only if n is an odd prime.
Order does not matter: a state containing one pile of two stones and one pile of three stones is considered the same as a state containing one pile of three stones and one pile of two stones.
A056219 is the analogous sequence when only one stone can be moved between piles, and A018819 is the analogous sequence when all stones must be moved between piles. - Peter Kagey, Sep 24 2017
Consider the inverse operation: either divide an even stack in two, or replace two stacks of equal parity with their averages. The states which have no inverse moves are precisely those with all parts equal and odd. If n is a power of two, its only odd divisor is 1 and so every state can be inverted to the all-ones state, and if n is an odd prime then the only states which cannot be inverted are the all-ones state and [n]. If n has an odd divisor d with 1 < d < n, then unreachable states include the all-d's state and any state obtainable from it by combining d's in pairs, of which there is at least one. Therefore, both conjectures are true. - Charlie Neder, Jan 28 2019

			For n = 5, the a(5) = 6 states are:
(1 1 1 1 1), (2 1 1 1), (2 2 1), (3 1 1), (3 2), and (4 1).
To reach state (3 2) starting from (1 1 1 1 1):
(1 1 1 1 1) -> (2 1 1 1) -> (2 2 1) -> (3 1 1) -> (3 2).

Charlie Neder, Table of n, a(n) for n = 1..60
Reddit User _AUTOMATIC_, Questions about this combinatorial puzzle

Cf. A000041, A018819, A056219.

a(44)-a(50) from Charlie Neder, Jan 28 2019

A307097 Number of configurations in the repeating cycle of the sandpile model in a bounded square of size 2n+1.

Original entry on oeis.org

2, 4, 4, 4, 6, 6, 8, 6, 10, 18, 26, 10, 10, 8, 12, 30, 8, 32, 14, 32, 52, 26, 60, 52, 34, 74, 14, 80, 36, 38, 24, 54, 26, 30, 36, 58, 22, 14, 26, 36, 38, 20, 36, 60, 24, 24, 18, 14, 24, 34, 70, 104, 48, 56, 36, 50, 50, 48, 152, 28, 110, 30, 172, 64, 104, 158, 150, 60, 36, 186, 52, 50

Offset: 1

Scott R. Shannon, Mar 24 2019

nonn

The Abelian sandpile model considers the behavior of grains of sand on a square grid when a square topples sand to its nearest neighbors when the number of grains in the square is greater than or equal to 4, where 4 is the number of nearest neighbors. If the grid is instead bounded by a square box then it is natural to extend this rule so that squares on the border also topple when they contain more or the same number of grains as the number of their nearest neighbors, i.e., 2 for corner squares, 3 for edge squares. Unlike the standard Abelian sandpile model on a finite grid, the square grid in this model does not lose sand, and assuming one keeps adding sand to the grid after each topple stabilization, eventually a critical number of grains will be added such that the resulting configurations will cycle. This sequence {a(n)} is the number of configurations in the cycle, for the sandpile model bounded by a square of size 2n+1, assuming the square grid starts with no sand and sand is continuously added to the center square until a cycle first occurs. As the toppling in a given area will repeat indefinitely, once the repeating cycle state is reached, one cannot topple the unstable squares in any random order and reach a stable configuration. To avoid such issues all squares are toppled simultaneously.

			For n=1 the square size is 2*1+1 = 3. The number of sand grains in each square is shown below, after the addition of 4, 8 and 12 grains to the central square:
.
   After 4         After 8         After 12
+---+---+---+   +---+---+---+   +---+---+---+
| 0 | 1 | 0 |   | 0 | 2 | 0 |   | 0 | 3 | 0 |
+---+---+---+   +---+---+---+   +---+---+---+
| 1 | 0 | 1 |   | 2 | 0 | 2 |   | 3 | 0 | 3 |
+---+---+---+   +---+---+---+   +---+---+---+
| 0 | 1 | 0 |   | 0 | 2 | 0 |   | 0 | 3 | 0 |
+---+---+---+   +---+---+---+   +---+---+---+
.
The edge squares now contain 3 grains, which means they are unstable as 3 equals their nearest neighbor count. This configuration thus topples to:
.
+---+---+---+
| 2 | 0 | 2 |
+---+---+---+
| 0 | 4 | 0 |
+---+---+---+
| 2 | 0 | 2 |
+---+---+---+
.
The central square is again unstable, as are the four corner squares as they contain 2 grains. This configuration topples to:
.
+---+---+---+
| 0 | 3 | 0 |
+---+---+---+
| 3 | 0 | 3 |
+---+---+---+
| 0 | 3 | 0 |
+---+---+---+
.
This is the same as the configuration after 12 grains above. These last two configurations cycle forever, thus a(1) = 2.

Scott R. Shannon, Table of n, a(n) for n = 1..500
Per Bak, Chao Tang, Kurt Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise, Phys. Rev. Lett. 59 (1987), 381-384.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Luis David Garcia-Puente and Brady Haran, Sandpiles, Numberphile video, YouTube.com, Jan. 13, 2017.
L. Levine, W. Pegden, C. K. Smart, Apollonian structure in the Abelian sandpile, Geom. Funct. Anal. (2016) 26:306.
Scott R. Shannon, Cycle for square of size 2*2+1 = 5. This cycle of 4 configurations is reached after the addition of 40 grains of sand. This is the first square size in a sequence of three sizes that have a cycle of 4: a(2), a(3), a(4). For this, and other images, white=0, yellow=1, green=2, blue=3, red=4, black>=5 grains.
Scott R. Shannon, Cycle for square of size 2*3+1 = 7. The second square to have a cycle of 4. This is reached after the addition of 100 grains.
Scott R. Shannon, Cycle for square of size 2*4+1 = 9. The third square to have a cycle of 4. This is reached after the addition of 160 grains.
Scott R. Shannon, Complete growth and cycle for square of size 2*7+1 = 15. This shows the addition of 468 grains (each image is pre-topple after the central square has regained 4 grains) followed by each iteration for the resulting 8-cycle configuration. The cycle is shown ten times.
Wikipedia, Abelian Sandpile Model.

Cf. A259013, A180230, A056219, A307652.

Cf. A007341 (order of the sandpile group of the (n-1)X(n-1) grid graph).

A166872 a(n) = floor(n/2 + 2 - sqrt(17/4 + 2*n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29

Offset: 1

Roger L. Bagula, Oct 22 2009

nonn

Each nonnegative integer k occurs 2 to 6 times in the sequence, with 0 occurring 6 times, 3 occurring 4 times, all others either 2 or 3 times. Conjecture: The sequence of integers k which occur 3 times has the g.f. (1 + 2*x^2 - 2*x^3 + x^6 - x^8 - x^9 + x^11)/(1 - 2*x + x^2 - x^8 + 2*x^9 - x^10). - Robert Israel, May 27 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Matthieu Latapy, Roberto Mantaci, Michel Morvan and Ha Duong Phan, Structure of some sand pile model, Theoretical Computer Science 262 (2001), pp. 525-556.

Cf. A056219.

nk:= k -> ceil(2*(k+1)+sqrt(16*k+17)) - ceil(2*k+sqrt(16*k+1)):
seq(k$nk(k), k=0..50); # Robert Israel, May 27 2016

Table[Floor[n/2 + 2 - Sqrt[2*n + 17/4]], {n, 1, 100}] (* G. C. Greubel, May 27 2016 *)

a(n) = floor(n/2 + 2 - sqrt(17/4 + 2*n)).

a(n) = k iff ceiling(2*k + sqrt(16*k+1)) <= n <= ceiling(2*k + 3 + sqrt(16*k+17)). - Robert Israel, May 27 2016

Edited by the associate editors of the OEIS, Nov 09 2009

cp's OEIS Frontend

A166869 a(n) = n * A056219(n+1).

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A166870 a(n) = n*(n-1)*A056219(n+1).

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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.

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A292726 a(n) is the number of states that can be achieved when starting from n piles each containing one stone, where any number of stones can be transferred between piles that start with the same number of stones.

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A307097 Number of configurations in the repeating cycle of the sandpile model in a bounded square of size 2n+1.

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A166872 a(n) = floor(n/2 + 2 - sqrt(17/4 + 2*n)).

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Maple

Mathematica

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A166870 a(n) = n(n-1)A056219(n+1).