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 A143785 #56 Sep 08 2022 08:45:37
%S A143785 3,8,20,36,63,96,144,200,275,360,468,588,735,896,1088,1296,1539,1800,
%T A143785 2100,2420,2783,3168,3600,4056,4563,5096,5684,6300,6975,7680,8448,
%U A143785 9248,10115,11016,11988,12996,14079,15200,16400,17640,18963,20328,21780,23276
%N A143785 Antidiagonal sums of the triangle A120070.
%C A143785 Let b(n) be the sequence (0,0,0,3,8,20,36,...), with offset 0. Then b(n) is the number of triples (w,x,y) having all terms in {0,...,n} and w < range{w,x,y}. - _Clark Kimberling_, Jun 11 2012
%C A143785 Consider a(n) with two 0's prepended and offset 1. Call the new sequence b(n) and consider the partitions of n into two parts (p,q). Then b(n) represents the sum of all the products (p + q) * (q - p) where p <= q. - _Wesley Ivan Hurt_, Apr 12 2018
%H A143785 Iain Fox, <a href="/A143785/b143785.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Colin Barker)
%H A143785 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A143785 a(n+1) - a(n) = A032438(n+2).
%F A143785 a(n) = A006918(n-2) + 2*A006918(n-1) + 3*A006918(n). - _R. J. Mathar_, Jul 01 2011
%F A143785 G.f.: x*(3+2*x+x^2) / ( (1+x)^2*(x-1)^4 ). - _R. J. Mathar_, Jul 01 2011
%F A143785 a(n) = (n+2)*(2*n^2 + 4*n - (-1)^n + 1)/8. - _Ilya Gutkovskiy_, May 07 2016
%F A143785 From _Colin Barker_, May 07 2016: (Start)
%F A143785 a(n) = (n^3 + 4*n^2 + 4*n)/4 for n even.
%F A143785 a(n) = (n^3 + 4*n^2 + 5*n + 2)/4 for n odd.
%F A143785 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 6. (End)
%F A143785 a(n) = Sum_{k=1..n+1} floor((n+1)*k/2). - _Wesley Ivan Hurt_, Apr 01 2017
%F A143785 a(n) = (n+2)*floor((n+1)^2/4) ( = (n+2)*A002620(n+1) ) for n > 0. - _Heinrich Ludwig_, Dec 22 2017
%F A143785 E.g.f.: e^(-x) * (-2 + x + e^(2*x)*(2 + 19*x + 14*x^2 + 2*x^3))/8. - _Iain Fox_, Dec 29 2017
%e A143785 First diagonal 3 = 3.
%e A143785 Second diagonal 8 = 8.
%e A143785 Third diagonal 5+15 = 20.
%e A143785 Fourth diagonal 24+12 = 36.
%t A143785 Rest@ CoefficientList[Series[x (3 + 2 x + x^2)/((1 + x)^2*(x - 1)^4), {x, 0, 44}], x] (* _Michael De Vlieger_, Dec 22 2017 *)
%t A143785 LinearRecurrence[{2, 1, -4, 1, 2, -1}, {3, 8, 20, 36, 63, 96}, 60] (* _Vincenzo Librandi_, Jan 22 2018 *)
%o A143785 (PARI) Vec(x*(3+2*x+x^2)/((1+x)^2*(x-1)^4) + O(x^50)) \\ _Colin Barker_, May 07 2016
%o A143785 (Magma) [(n+2)*(2*n^2+4*n-(-1)^n+1)/8: n in [1..50]]; // _Vincenzo Librandi_, Jan 22 2018
%Y A143785 Cf. A035006, A099721 (bisections).
%Y A143785 Cf. A120070, A002620.
%K A143785 nonn,easy
%O A143785 1,1
%A A143785 _Paul Curtz_, Sep 01 2008