This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103220 #74 Sep 22 2023 05:18:47
%S A103220 0,1,13,58,170,395,791,1428,2388,3765,5665,8206,11518,15743,21035,
%T A103220 27560,35496,45033,56373,69730,85330,103411,124223,148028,175100,
%U A103220 205725,240201,278838,321958,369895,422995,481616,546128,616913,694365,778890
%N A103220 a(n) = n*(n+1)*(3*n^2+n-1)/6.
%C A103220 Row sums of A103219.
%C A103220 From _Bruno Berselli_, Dec 10 2010: (Start)
%C A103220 a(n) = n*A002412(n) - Sum_{i=0..n-1} A002412(i). More generally: n^2*(n+1)*(2*d*n-2*d+3)/6 - (Sum_{i=0..n-1} i*(i+1)*(2*d*i-2*d+3))/6 = n * (n+1) * (3*d*n^2-d*n+4*n-2*d+2)/12; in this sequence is d=2.
%C A103220 The inverse binomial transform yields 0, 1, 11, 22, 12, 0, 0 (0 continued). (End)
%C A103220 a(n-1) is also number of ways to place 2 nonattacking semi-queens (see A099152) on an n X n board. - _Vaclav Kotesovec_, Dec 22 2011
%C A103220 Also, one-half the even-indexed terms of the partial sums of A045947. - _J. M. Bergot_, Apr 12 2018
%H A103220 Vincenzo Librandi, <a href="/A103220/b103220.txt">Table of n, a(n) for n = 0..1000</a>
%H A103220 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A103220 G.f.: x*(1+8*x+3*x^2)/(1-x)^5.
%F A103220 a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j)^2. - _Enrique Pérez Herrero_, Jan 15 2013
%F A103220 a(n) = a(n-1) + (2*n-1)*n^2 with a(0)=0, see A015237. - _J. M. Bergot_, Jun 10 2017
%F A103220 From _Wesley Ivan Hurt_, Nov 20 2021: (Start)
%F A103220 a(n) = Sum_{k=1..n} k * C(2*k,2).
%F A103220 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). (End)
%F A103220 From _Peter Bala_, Sep 03 2023: (Start)
%F A103220 a(n) = Sum_{1 <= i <= j <= n} (2*i - 1)*(2*j - 1).
%F A103220 Second subdiagonal of A039755. (End)
%p A103220 for(n=0,100,print1((3*n^4+4*n^3-n)/6,","))
%t A103220 CoefficientList[Series[- x (1 + 8 x + 3 x^2) / (x - 1)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 12 2013 *)
%t A103220 LinearRecurrence[{5,-10,10,-5,1},{0,1,13,58,170},40] (* _Harvey P. Dale_, Jan 23 2016 *)
%o A103220 (PARI) a(n)=n*(n+1)*(3*n^2+n-1)/6 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A103220 Cf. A002412, A002418, A039755, A099152, A103219, A015237 (first diffs), A024196.
%K A103220 easy,nonn
%O A103220 0,3
%A A103220 Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 25 2005