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 A386706 #29 Aug 20 2025 10:57:30
%S A386706 0,1,5,18,71,260,990,3672,13775,51343,191860,715770,2672298,9972092,
%T A386706 37220040,138903480,518408351,1934712530,7220497115,26947209762,
%U A386706 100568547820,375326739216,1400739172470,5227629044040,19509779871450,72811487038701,271736178975820,1014133216234068
%N A386706 Expansion of ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.
%C A386706 a(n) is the value at q = 2 + sqrt(3) of C_n(q)/(q^{n-1}(q - 1)^2), where C_n(q) is the number of codimension n ideals of the algebra of two-variable Laurent polynomials over a finite field of order q. The number C_n(q) is a palindromic polynomial of degree 2n with integer coefficients in the variable q and it is divisible by (q-1)^2.
%H A386706 Vaclav Kotesovec, <a href="/A386706/b386706.txt">Table of n, a(n) for n = 0..1000</a>
%H A386706 Christian Kassel and Christophe Reutenauer, <a href="http://arxiv.org/abs/1505.072294">Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables</a>, arXiv:1505.07229 [math.AG], 2015-2016; Michigan Math. J. 67 (2018), 715-741.
%H A386706 Christian Kassel and Christophe Reutenauer, <a href="http://arxiv.org/abs/1610.07793">Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus</a>, arXiv:1610.07793 [math.NT], 2016; The Ramanujan Journal 46 (2018), 633-655.
%H A386706 Christian Kassel and Christophe Reutenauer, <a href="http://arxiv.org/abs/2507.15780">Pairs of intertwined integer sequences</a>, arXiv:2507.15780 [math.NT], 2025.
%F A386706 G.f.: ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.
%F A386706 a(2^k) = A001834(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 of Kassel-Reutenauer paper "Pairs of intertwined integer sequences".
%F A386706 a(n) ~ (1 + sqrt(3))^(2*n-1) / 2^n. - _Vaclav Kotesovec_, Jul 30 2025
%t A386706 nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^2/(1 - 4*x^k + x^(2*k)), {k, 1, nmax}] - 1)/2, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%Y A386706 Cf. A001834, A329156, A387017.
%K A386706 nonn
%O A386706 0,3
%A A386706 _Christian Kassel_, Jul 30 2025
%E A386706 a(0)=0 added, offset changed to 0, a(7) corrected and more terms added by _Vaclav Kotesovec_, Jul 30 2025