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 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 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 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(n-1, 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

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Last modified December 6 14:47 EST 2019. Contains 329806 sequences. (Running on oeis4.)