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 A275709 #100 Jan 29 2025 13:15:15
%S A275709 0,5,28,81,176,325,540,833,1216,1701,2300,3025,3888,4901,6076,7425,
%T A275709 8960,10693,12636,14801,17200,19845,22748,25921,29376,33125,37180,
%U A275709 41553,46256,51301,56700,62465,68608,75141,82076,89425,97200,105413,114076,123201,132800,142885
%N A275709 a(n) = 2*n^3 + 3*n^2.
%C A275709 Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link).
%C A275709 a(n) is the maximal determinant of a 3 X 3 matrix with integer elements from {1, ..., n+1}, so (for example) the maximum determinant of a 3 X 3 matrix with integer elements from {1, ..., 5} = det(1, 5, 5; 5, 1, 5; 5, 5, 1) = a(4) = 176. - _Matthew Scroggs_, Dec 31 2022
%H A275709 Carauleanu Marc and Colin Barker, <a href="/A275709/b275709.txt">Table of n, a(n) for n = 0..3030</a> (first 1000 terms from Colin Barker)
%H A275709 Joshua Giambalvo, <a href="http://imgur.com/gallery/2BeBR">Illustration of initial terms in a square array</a>, Imgur, (2016).
%H A275709 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A275709 From _Colin Barker_, Aug 06 2016: (Start)
%F A275709 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
%F A275709 G.f.: x*(5+8*x-x^2) / (1-x)^4. (End)
%F A275709 a(n) = A033431(n) + A033428(n). - _Omar E. Pol_, Aug 09 2016
%F A275709 a(n) = A000290(n) * A005408(n+1). - _Robert Israel_, Aug 09 2016
%F A275709 a(n) = A320047(1, n, 0). - _Kolosov Petro_, Oct 04 2018
%F A275709 E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - _G. C. Greubel_, Oct 19 2018
%F A275709 From _Amiram Eldar_, Jan 21 2023: (Start)
%F A275709 Sum_{n>=1} 1/a(n) = Pi^2/18 + 4*log(2)/9 - 16/27.
%F A275709 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 + Pi/9 -2*log(2)/9 - 8/27. (End)
%p A275709 seq(2*n^3+3*n^2, n=0..30); # _Robert Israel_, Aug 09 2016
%t A275709 Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *)
%t A275709 CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* _Michael De Vlieger_, Aug 11 2016 *)
%o A275709 (PARI) concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ _Colin Barker_, Aug 28 2016
%o A275709 (PARI) a(n)=n^2*(2*n+3) \\ _Charles R Greathouse IV_, Aug 28 2016
%o A275709 (Magma) [n^2*(2*n + 3): n in [0..30]]; // _G. C. Greubel_, Oct 19 2018
%o A275709 (Python) for n in range(0,50): print(n**2*(2*n+3), end=' ') # _Stefano Spezia_, Oct 19 2018
%Y A275709 Cf. A000290, A005408, A005563, A033428, A033431, A057145, A139600.
%K A275709 nonn,easy
%O A275709 0,2
%A A275709 _Joshua Giambalvo_, Aug 06 2016