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 A255211 #52 Mar 02 2025 10:03:30
%S A255211 0,3,16,46,100,185,308,476,696,975,1320,1738,2236,2821,3500,4280,5168,
%T A255211 6171,7296,8550,9940,11473,13156,14996,17000,19175,21528,24066,26796,
%U A255211 29725,32860,36208,39776,43571,47600,51870,56388,61161,66196,71500,77080,82943
%N A255211 a(n) = n*(n+1)*(7*n+2)/6.
%C A255211 a(n) is the number of triangles of all sizes in a polyiamond of trapezoid shape with 3 sides of length n and the base of length 2*n. The number of triangular cells in the trapezoid is 3*n^2. This is half of a regular hexagon with side lengths n.
%C A255211 The number of triangles oriented with their bases aligned with the base of the trapezoid is n*(n+1)*(2*n+1)/3 and the number oriented in the opposite direction is n^2*(n+1)/2. a(n) is the sum of these two.
%H A255211 Colin Barker, <a href="/A255211/b255211.txt">Table of n, a(n) for n = 0..1000</a>
%H A255211 Luce ETIENNE, <a href="/A255211/a255211.pdf">Illustration a(1), a(2), a(3), a(4) and a(5)</a>
%H A255211 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A255211 G.f.: x*(3 + 4*x) / (1 - x)^4. - _Colin Barker_, Feb 17 2015
%F A255211 a(n) = Sum_{j=0..n-1} (n-j)*(3*n-2*j) = Sum_{j=1..n} j*(n+2*j) for n>0.
%F A255211 a(n) = A000292(2*n) - A000292(n). - _Bruno Berselli_, Sep 22 2016
%F A255211 Sum_{n>=1} 1/a(n) = 21*HarmonicNumber(2/7)/5 - 6/5 = 0.44513027538601361333... . - _Vaclav Kotesovec_, Sep 22 2016
%F A255211 E.g.f.: exp(x)*x*(18 + 30*x + 7*x^2)/6. - _Stefano Spezia_, Mar 02 2025
%e A255211 From the second comment: a(1)= 2+1, a(2)= 10+6, a(3)= 28+18, a(4)= 60+40.
%t A255211 Table[n (n + 1) (7 n + 2)/6, {n, 0, 50}] (* _Bruno Berselli_, Feb 17 2015 *)
%o A255211 (PARI) concat(0, Vec(x*(4*x+3)/(x-1)^4 + O(x^100))) \\ _Colin Barker_, Feb 17 2015
%o A255211 (PARI) vector(50, n, n--; n*(n+1)*(7*n+2)/6) \\ _Bruno Berselli_, Feb 17 2015
%o A255211 (Magma) [n*(n+1)*(7*n+2)/6 : n in [0..50]]; // _Wesley Ivan Hurt_, Apr 11 2021
%Y A255211 Partial sums of A022264.
%Y A255211 Cf. A000292, A000330, A006331, A002411, A033428, A212977.
%K A255211 nonn,easy
%O A255211 0,2
%A A255211 _Luce ETIENNE_, Feb 17 2015
%E A255211 Edited and extended by _Bruno Berselli_, Dec 01 2016