A307172 Second class of all proper positive solutions x2(n) of the Pell equation x^2 - 7*y^2 = 9.

4, 53, 844, 13451, 214372, 3416501, 54449644, 867777803, 13829995204, 220412145461, 3512764332172, 55983817169291, 892228310376484, 14219669148854453, 226622478071294764, 3611739979991861771, 57561217201798493572, 917367735248784035381, 14620322546778746072524

Offset: 1

Author

Wolfdieter Lang, Mar 27 2019

Keywords

• nonn
• easy

Comments

The corresponding y solutions are y2(n) = A307173(n).

See A307168 for details.

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).

Links

• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (16,-1).

Crossrefs

Cf. A077421, A307168, A307169, A307173.

Formula

G.f.: x*(4 - 11*x)/(1 - 16*x + x^2).

a(n) = 11*S(n, 16) - 172*S(n-1, 16) for n >= 1, with S(n, 16) = A077412(n).

a(n) = sqrt(9 + 7*A307173(n)^2) for n >= 1.