cp's OEIS Frontend

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

%I A188623 #30 Apr 06 2025 08:46:22
%S A188623 1,2,2,5,7,8,12,15,17,22,26,29,35,40,44,51,57,62,70,77,83,92,100,107,
%T A188623 117,126,134,145,155,164,176,187,197,210,222,233,247,260,272,287,301,
%U A188623 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
%N A188623 Number of reachable configurations in a chip-firing game on a triangle starting with n chips on one vertex.
%C A188623 Quasipolynomial with period 3 (see formulas below).
%H A188623 Yifan Xie, <a href="/A188623/b188623.txt">Table of n, a(n) for n = 1..10000</a>
%H A188623 J. Schneider, <a href="http://arxiv.org/abs/1104.0279v1">Enumeration and Quasipolynomiality of Chip-Firing Configurations</a>, arXiv:1104.0279 [math.CO], 2011.
%H A188623 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A188623 a(3*k)   = (3*k^2 + 3*k - 2)/2,
%F A188623 a(3*k+1) = (3*k^2 + 5*k + 2)/2,
%F A188623 a(3*k+2) = (3*k^2 + 7*k + 4)/2.
%F A188623 G.f.: x*(1 - x^2 + 2*x^3 - x^4)/((1 + x + x^2)*(1 - x)^3). - _Bruno Berselli_, Feb 03 2016
%F A188623 a(n) = (n*(n + 3) - 4*(-1)^floor(2*n/3 + 1/3) - 2)/6. - _Bruno Berselli_, Feb 03 2016
%e A188623 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).
%t A188623 Table[(n (n + 3) - 4 (-1)^Floor[2 n/3 + 1/3] - 2)/6, {n, 1, 80}]
%t A188623 (* _Bruno Berselli_, Feb 03 2016 *)
%o A188623 (Sage) [(n*(n+3)-4*(-1)^floor(2*n/3+1/3)-2)/6 for n in (1..80)] # _Bruno Berselli_, Feb 03 2016
%K A188623 nonn,easy
%O A188623 1,2
%A A188623 _Jon Schneider_, Apr 05 2011