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 A307172 #16 May 31 2022 14:29:29
%S A307172 4,53,844,13451,214372,3416501,54449644,867777803,13829995204,
%T A307172 220412145461,3512764332172,55983817169291,892228310376484,
%U A307172 14219669148854453,226622478071294764,3611739979991861771,57561217201798493572,917367735248784035381,14620322546778746072524
%N A307172 Second class of all proper positive solutions x2(n) of the Pell equation x^2 - 7*y^2 = 9.
%C A307172 The corresponding y solutions are y2(n) = A307173(n).
%C A307172 See A307168 for details.
%C A307172 The proper positive solutions (x2(n), y2(n)) are given in matrix notation by R(0)*R(2)*Auto(n)*R^{-1}(4)*R^{-1}(-1)*R^{-1}(3)*(1, 0)^T (T for transposed), with the R-matrix R(t) = Matrix([[0, -1],[1, t]]), its inverse R^{-1}(t) = Matrix([t, 1],[-1, 0]), and the automorphic matrix Auto = Matrix([2, 9],[3, 14]). The matrix power Auto^n is given in A307168 in terms of Chebyshev polynomials S(n, x=16) = A077412(n).
%H A307172 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A307172 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A307172 G.f.: x*(4 - 11*x)/(1 - 16*x + x^2).
%F A307172 a(n) = 11*S(n, 16) - 172*S(n-1, 16) for n >= 1, with S(n, 16) = A077412(n).
%F A307172 a(n) = sqrt(9 + 7*A307173(n)^2) for n >= 1.
%Y A307172 Cf. A077421, A307168, A307169, A307173.
%K A307172 nonn,easy
%O A307172 1,1
%A A307172 _Wolfdieter Lang_, Mar 27 2019