A188623 Number of reachable configurations in a chip-firing game on a triangle starting with n chips on one vertex.

1, 2, 2, 5, 7, 8, 12, 15, 17, 22, 26, 29, 35, 40, 44, 51, 57, 62, 70, 77, 83, 92, 100, 107, 117, 126, 134, 145, 155, 164, 176, 187, 197, 210, 222, 233, 247, 260, 272, 287, 301, 314, 330, 345, 359, 376, 392, 407, 425, 442, 458, 477, 495, 512, 532, 551, 569, 590, 610, 629, 651, 672, 692, 715, 737, 758, 782, 805, 827, 852, 876, 899, 925, 950, 974, 1001

Offset

1,2

Comments

Quasipolynomial with period 3 (see formulas below).

Links

Table of n, a(n) for n=1..76.

J. Schneider, Enumeration and Quasipolynomiality of Chip-Firing Configurations, arXiv:1104.0279 [math.CO], 2011.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(3*k) = (3*k^2 + 3*k - 2)/2,

a(3*k+1) = (3*k^2 + 5*k + 2)/2,

a(3*k+2) = (3*k^2 + 7*k + 4)/2.

G.f.: x*(1 - x^2 + 2*x^3 - x^4)/((1 + x + x^2)*(1 - x)^3). [Bruno Berselli, Feb 03 2016]

a(n) = (n*(n + 3) - 4*(-1)^floor(2*n/3 + 1/3) - 2)/6. [Bruno Berselli, Feb 03 2016]

Example

For n=4, a(4)=5 because the reachable configurations are: (4, 0, 0), (2, 1, 1), (0, 2, 2), (1, 0, 3), (3, 0, 1).

Mathematica

Table[(n (n + 3) - 4 (-1)^Floor[2 n/3 + 1/3] - 2)/6, {n, 1, 80}]
(* Bruno Berselli, Feb 03 2016 *)

Prog

(Sage) [(n*(n+3)-4*(-1)^floor(2*n/3+1/3)-2)/6 for n in (1..80)] # Bruno Berselli, Feb 03 2016

Crossrefs

Sequence in context: A216392 A287908 A239259 * A256358 A308842 A241761

Adjacent sequences: A188620 A188621 A188622 * A188624 A188625 A188626

Keyword

nonn,easy

Author

Jon Schneider, Apr 05 2011

Status

approved