

A307172


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


2



4, 53, 844, 13451, 214372, 3416501, 54449644, 867777803, 13829995204, 220412145461, 3512764332172, 55983817169291, 892228310376484, 14219669148854453, 226622478071294764, 3611739979991861771, 57561217201798493572, 917367735248784035381, 14620322546778746072524
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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 Rmatrix 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

Table of n, a(n) for n=1..19.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,1).


FORMULA

G.f.: x*(4  11*x)/(1  16*x + x^2).
a(n) = 11*S(n, 16)  172*S(n1, 16) for n >= 1, with S(n, 16) = A077412(n).
a(n) = sqrt(9 + 7*A07173(n)) for n >= 1.


CROSSREFS

Cf. A077421, A307168, A307169, A307173.
Sequence in context: A109801 A099340 A221605 * A275801 A158259 A095210
Adjacent sequences: A307169 A307170 A307171 * A307173 A307174 A307175


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Mar 27 2019


STATUS

approved



