|
%I #21 Apr 28 2023 17:45:11
%S 1,0,4,29,107,286,630,1219,2149,3532,5496,8185,11759,16394,22282,
%T 29631,38665,49624,62764,78357,96691,118070,142814,171259,203757,
%U 240676,282400,329329,381879,440482,505586,577655,657169,744624,840532,945421,1059835,1184334,1319494
%N a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1.
%C For n >=1, a(n) is the number of divisions of a 2 X n board into 3 pieces. See Jacob Brown article.
%H Michael De Vlieger, <a href="/A345897/b345897.txt">Table of n, a(n) for n = 0..10000</a>
%H Jacob Brown, <a href="https://arxiv.org/abs/2106.14755">Counting Divisions of a 2 X n Rectangular Grid</a>, arXiv:2106.14755 [math.CO], 2021.
%H Tomislav Došlić and Luka Podrug, <a href="https://arxiv.org/abs/2304.12121">Sweet division problems: from chocolate bars to honeycomb strips and back</a>, arXiv:2304.12121 [math.CO], 2023.
%H Samuel Durham and Tom Richmond, <a href="https://doi.org/10.1080/07468342.2020.1674597">Connected Subsets of an n X 2 Rectangle</a>, The College Mathematics Journal, Volume 51, 2020 - Issue 1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F From _Chai Wah Wu_, Jun 29 2021: (Start)
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
%F G.f.: (1 - 5*x + 14*x^2 - x^3 + 7*x^4)/(1-x)^5. (End)
%t CoefficientList[Series[(1 - 5 x + 14 x^2 - x^3 + 7 x^4)/(1 - x)^5, {x, 0, 38}], x] (* _Michael De Vlieger_, Apr 28 2023 *)
%o (PARI) a(n) = 2*n^4/3 - 4*n^3/3 + 11*n^2/6 - 13*n/6 + 1;
%Y Cf. A172482 (same but where the rightmost squares separate).
%K nonn,easy
%O 0,3
%A _Michel Marcus_, Jun 29 2021
|
