cp's OEIS Frontend

A355645 The number of regions in the G-Shi arrangement when G is the cycle graph C_n.

%I A355645 #30 May 24 2024 16:26:11
%S A355645 1,3,16,61,206,659,2052,6297,19162,58015,175088,527333,1586118,
%T A355645 4766571,14316124,42981169,129009074,387158327,1161737160,3485735805,
%U A355645 10458256030,31376865283,94134790196,282412759241,847255054986,2541798719439,7625463267232
%N A355645 The number of regions in the G-Shi arrangement when G is the cycle graph C_n.
%C A355645 The G-Shi arrangement of a graph G is the hyperplane arrangement given by hyperplanes x_i - x_j = 0 and x_i - x_j = 1 for each edge {i,j} of G with i < j.
%C A355645 a(n) is also the number of parking functions of length n with all cars preferring to park in the first 3 spots. - _Jayden Thadani_, May 20 2024
%H A355645 Paolo Xausa, <a href="/A355645/b355645.txt">Table of n, a(n) for n = 1..1000</a>
%H A355645 Douglas M. Chen, <a href="https://arxiv.org/abs/2311.15699">On the Structure of Permutation Invariant Parking</a>, arXiv:2311.15699 [math.CO], 2023. See p. 16.
%H A355645 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,17,-6).
%F A355645 a(n) = 3^n - 2^n - n for n >= 2.
%F A355645 G.f.: x*(1 - 4*x + 12*x^2 - 17*x^3 + 6*x^4)/((1 - x)^2*(1 - 2*x)*(1 - 3*x)). - _Stefano Spezia_, Jul 12 2022
%t A355645 A355645[n_] := If[n == 1, 1, 3^n - 2^n - n]; Array[A355645, 50] (* or *)
%t A355645 LinearRecurrence[{7, -17, 17, -6}, {1, 3, 16, 61, 206}, 50] (* _Paolo Xausa_, May 24 2024 *)
%Y A355645 Cf. A004146 (cycle graph labels).
%Y A355645 Cf. A000244 (path graph regions), A001906 (path graph labels).
%K A355645 nonn,easy
%O A355645 1,2
%A A355645 _Robin Truax_, Jul 11 2022