login
a(n) = (1+n)*(9 + 11*n + 4*n^2)/3.
5

%I #26 Sep 08 2022 08:45:50

%S 3,16,47,104,195,328,511,752,1059,1440,1903,2456,3107,3864,4735,5728,

%T 6851,8112,9519,11080,12803,14696,16767,19024,21475,24128,26991,30072,

%U 33379,36920,40703,44736,49027,53584,58415,63528,68931,74632,80639,86960,93603

%N a(n) = (1+n)*(9 + 11*n + 4*n^2)/3.

%C One of the bisections of the left central column in the Janet table A172002.

%C Row 1 of the convolution array A213844. - _Clark Kimberling_, Jul 05 2012

%C With offset 2, this is 4*n^3/3 - 3*n^2 + 8*n/3 - 1, the number of divisions of a 2 X n board into 3 pieces where the rightmost squares separate. See Jacob Brown article. - _Michel Marcus_, Jun 29 2021

%H Vincenzo Librandi, <a href="/A172482/b172482.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×n Rectangular Grid</a>, arXiv:2106.14755 [math.CO], 2021.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = A131941(2n+2), where A100178(n) = A131941(2n-1).

%F a(n) = 4*a(n) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F a(n) mod 10 = 3, 6, 7, 4, 5, 8, 1, 2, 9, 0 (and repeat periodically).

%F G.f.: (x+3)*(1+x)/(x-1)^4.

%t CoefficientList[Series[(x + 3) (1 + x)/(x - 1)^4, {x, 0, 40}], x] (* _Michael De Vlieger_, Nov 02 2018 *)

%o (Magma) [(1+n)*(9+11*n+4*n^2)/3: n in [0..40]]; // _Vincenzo Librandi_, Aug 04 2011

%o (PARI) a(n)=(1+n)*(9+11*n+4*n^2)/3 \\ _Charles R Greathouse IV_, Aug 04 2011

%Y Cf. A100178, A131941.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Feb 04 2010

%E Edited by _R. J. Mathar_, Feb 24 2010