OFFSET
0,3
COMMENTS
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.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Christian Kassel and Christophe Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, arXiv:1505.07229 [math.AG], 2015-2016; Michigan Math. J. 67 (2018), 715-741.
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016; The Ramanujan Journal 46 (2018), 633-655.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025.
FORMULA
G.f.: ((Product_{k>=1} (1 - x^k)^2/(1 - 4*x^k + x^(2k))) - 1)/2.
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".
a(n) ~ (1 + sqrt(3))^(2*n-1) / 2^n. - Vaclav Kotesovec, Jul 30 2025
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Christian Kassel, Jul 30 2025
EXTENSIONS
a(0)=0 added, offset changed to 0, a(7) corrected and more terms added by Vaclav Kotesovec, Jul 30 2025
STATUS
approved