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 A385029 #27 Jun 18 2025 12:24:44
%S A385029 18,250,1372,4860,13310,30758,63000,117912,205770,339570,535348,
%T A385029 812500,1194102,1707230,2383280,3258288,4373250,5774442,7513740,
%U A385029 9648940,12244078,15369750,19103432,23529800,28741050,34837218,41926500,50125572,59559910,70364110,82682208,96668000
%N A385029 a(n) = Sum_{-n <= a, b, c <= n} (b^2 - 4*a*c).
%C A385029 There are (2*n + 1)^3 combinations of a, b, c.
%H A385029 Paolo Xausa, <a href="/A385029/b385029.txt">Table of n, a(n) for n = 1..10000</a>
%H A385029 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A385029 a(n) = (n*(n+1)*(2*n+1)^3)/3.
%F A385029 a(n) = (A055112(n)*A016754(n))/3.
%F A385029 a(n) = (A002378(n)*A016755(n))/3.
%F A385029 G.f.: 2*x*(9 + 71*x + 71*x^2 + 9*x^3)/(1 - x)^6. - _Stefano Spezia_, Jun 15 2025
%F A385029 From _Amiram Eldar_, Jun 18 2025; (Start)
%F A385029 Sum_{n>=1} 1/a(n) = 21*(1 - zeta(3)/2) - 12*log(2).
%F A385029 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi^3/8 + 3*Pi - 21. (End)
%t A385029 A385029[n_] := (n*(n + 1)*(2*n + 1)^3)/3;
%t A385029 Array[A385029, 50] (* _Paolo Xausa_, Jun 18 2025 *)
%o A385029 (Python)
%o A385029 a = lambda n: ((n*n+n)*((n << 1)+1)**3)//3
%o A385029 print([a(n) for n in range(1, 11)])
%Y A385029 Cf. A000384, A002378, A014105, A016755, A016754, A055112, A384666.
%K A385029 nonn,easy
%O A385029 1,1
%A A385029 _Darío Clavijo_, Jun 15 2025