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 A270710 #40 Feb 16 2025 08:33:32
%S A270710 -1,4,15,32,55,84,119,160,207,260,319,384,455,532,615,704,799,900,
%T A270710 1007,1120,1239,1364,1495,1632,1775,1924,2079,2240,2407,2580,2759,
%U A270710 2944,3135,3332,3535,3744,3959,4180,4407,4640,4879,5124,5375,5632,5895,6164,6439,6720,7007,7300,7599
%N A270710 a(n) = 3*n^2 + 2*n - 1.
%C A270710 In general, the ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k, is (k + (p + q - 2*k)*x + (p - q + k)*x^2)/(1 - x)^3.
%C A270710 From _Bruno Berselli_, Mar 25 2016: (Start)
%C A270710 This sequence and A140676 provide all integer m such that 3*m + 4 is a square.
%C A270710 Numbers related to A135713 by A135713(n) = n*a(n) - Sum_{k=0..n-1} a(k).
%C A270710 After -1, second bisection of A184005. (End)
%H A270710 Ilya Gutkovskiy, <a href="/A270710/a270710.pdf">Examples of the ordinary generating function for the values of quadratic polynomial</a>.
%H A270710 Leo Tavares, <a href="/A270710/a270710.jpg">Triple Diamond Illustration</a>.
%H A270710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuadraticPolynomial.html">Quadratic Polynomial</a>.
%H A270710 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A270710 G.f.: (-1 + 7*x)/(1 - x)^3.
%F A270710 E.g.f.: exp(x)*(-1 + 5*x + 3*x^2).
%F A270710 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A270710 a(n) = A033428(n) + A060747(n).
%F A270710 a(n) = A045944(n) - 1 = A056109(n) - 2.
%F A270710 a(-n) = A140676(n-1), with A140676(-1) = -1.
%F A270710 Sum_{n>=0} 1/a(n) = 3*(log(3) - 2)/8 - Pi/(8*sqrt(3)) = -0.564745312278736...
%F A270710 a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - _Bruno Berselli_, Feb 16 2018
%F A270710 a(n) = A000290(n+1) + 2*A000290(n) - 2. - _Leo Tavares_, May 28 2023
%F A270710 Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + 3/4. - _Amiram Eldar_, Jul 20 2023
%e A270710 a(0) = 3*0^2 + 2*0 - 1 = -1;
%e A270710 a(1) = 3*1^2 + 2*1 - 1 = 4;
%e A270710 a(2) = 3*2^2 + 2*2 - 1 = 15;
%e A270710 a(3) = 3*3^2 + 2*3 - 1 = 32, etc.
%t A270710 Table[3 n^2 + 2 n - 1, {n, 0, 50}]
%t A270710 LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]
%o A270710 (PARI) Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ _Michel Marcus_, Mar 22 2016
%o A270710 (PARI) lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ _Altug Alkan_, Mar 25 2016
%o A270710 (PARI) vector(50, n, n--; 3*n^2+2*n-1) \\ _Bruno Berselli_, Mar 25 2016
%o A270710 (Sage) [3*n^2+2*n-1 for n in (0..50)] # _Bruno Berselli_, Mar 25 2016
%o A270710 (Maxima) makelist(3*n^2+2*n-1, n, 0, 50); /* _Bruno Berselli_, Mar 25 2016 */
%o A270710 (Magma) [3*n^2+2*n-1: n in [0..50]]; // _Bruno Berselli_, Mar 25 2016
%o A270710 (GAP) List([0..50], n -> 3*n^2+2*n-1); # _Bruno Berselli_, Feb 16 2018
%Y A270710 Cf. A033428, A045944, A056105, A056109, A060747, A135713, A140676, A184005.
%Y A270710 Cf. A000290.
%K A270710 sign,easy
%O A270710 0,2
%A A270710 _Ilya Gutkovskiy_, Mar 22 2016