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A213840 a(n) = n*(1 + n)*(3 - 4*n + 4*n^2)/6.

%I A213840 #64 Sep 08 2022 08:46:02
%S A213840 1,11,54,170,415,861,1596,2724,4365,6655,9746,13806,19019,25585,33720,
%T A213840 43656,55641,69939,86830,106610,129591,156101,186484,221100,260325,
%U A213840 304551,354186,409654,471395,539865,615536,698896,790449,890715,1000230,1119546,1249231
%N A213840 a(n) = n*(1 + n)*(3 - 4*n + 4*n^2)/6.
%C A213840 Antidiagonal sums of the convolution array A213838.
%C A213840 The sequence is the binomial transform of (1, 10, 33, 40, 16, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 31 2015
%C A213840 From _Mircea Dan Rus_, Jul 11 2020: (Start)
%C A213840 a(n) is also the number of rectangles in a square biscuit of order n, which is obtained by stacking 2n-1 rows with their centers vertically aligned which consist successively of 1, 3, ..., 2n-3, 2n-1, 2n-3, ..., 3, 1 consecutive unit lattice squares. The order 2 and 3 square biscuits are shown below which contain 11 and 54 rectangles respectively.
%C A213840                         __
%C A213840         __           __|__|__
%C A213840      __|__|__     __|__|__|__|__
%C A213840     |__|__|__|   |__|__|__|__|__|
%C A213840        |__|         |__|__|__|
%C A213840                        |__|
%C A213840 (End)
%H A213840 Clark Kimberling, <a href="/A213840/b213840.txt">Table of n, a(n) for n = 1..200</a>
%H A213840 Teofil Bogdan and Mircea Rus, <a href="https://ssmr.ro/gazeta/gmb/2020/6/articol.pdf">Numărând dreptunghiuri pe foaia de matematică</a> (in Romanian). Gazeta Matematică, seria B, 2020 (6-7-8), pp. 281-288.
%H A213840 Teofil Bogdan and Mircea Dan Rus, <a href="https://arxiv.org/abs/2007.13472">Counting the lattice rectangles inside Aztec diamonds and square biscuits</a>, arXiv:2007.13472 [math.CO], 2020.
%H A213840 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A213840 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A213840 G.f.: x*(1 + 3*x)^2/(1 - x)^5.
%F A213840 From _Mircea Dan Rus_, Aug 26 2020: (Start)
%F A213840 a(n) = A000332(n+3) + 6*A000332(n+2) + 9*A000332(n+1).
%F A213840 a(n) = A002417(n) + 3*A002417(n-1). (End)
%p A213840 A213840:=n->n*(1 + n)*(3 - 4*n + 4*n^2)/6: seq(A213840(n), n=1..50); # _Wesley Ivan Hurt_, Sep 16 2017
%t A213840 Table[n (1 + n) (3 - 4 n + 4 n^2)/6, {n, 50}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 11, 54, 170, 415}, 40] (* _Vincenzo Librandi_, Aug 01 2015 *)
%o A213840 (Magma) [n*(1+n)*(3-4*n+4*n^2)/6: n in [1..60]]; // _Vincenzo Librandi_, Aug 01 2015
%Y A213840 Cf. A000332, A002417, A213838.
%Y A213840 First differences of A271870. - _J. M. Bergot_, Aug 29 2016
%K A213840 nonn,easy
%O A213840 1,2
%A A213840 _Clark Kimberling_, Jul 05 2012
%E A213840 Edited (with simpler definition) by _N. J. A. Sloane_, Sep 19 2017