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A188623 Number of reachable configurations in a chip-firing game on a triangle starting with n chips on one vertex. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A241761 A278388

Adjacent sequences:  A188620 A188621 A188622 * A188624 A188625 A188626

KEYWORD

nonn,easy

AUTHOR

Jon Schneider, Apr 05 2011

STATUS

approved

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)