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%I #24 Sep 08 2022 08:46:11
%S 5,31,181,1055,6149,35839,208885,1217471,7095941,41358175,241053109,
%T 1404960479,8188709765,47727298111,278175078901,1621323175295,
%U 9449763972869,55077260661919,321013799998645,1871005539329951,10905019435981061,63559111076556415
%N All positive solutions x of the second class of the Pell equation x^2 - 2*y^2 = -7.
%C For the corresponding y = y2 terms see 2*A038725(n+1).
%C The Pell equation x^2 - 2*y^2 = 7 has two classes of solutions. See, e.g., the Nagell reference and comments under A254938 and A255233. Here the positive solutions based on the fundamental solution (5, 4) (the second largest positive solution) are considered.
%C The positive solutions of the first class are given in (A054490(n), 2*A038723(n)), n >= 0.
%C The combined solutions of both classes are given in (A077446, 4*A077447).
%C The solutions (x(n), y(n)) of x^2 - 2*y^2 = -7 translate to the solutions (X(n), Y(n)) = (2*y(n) , x(n)) of the Pell equation X^2 - 2*Y^2 = 14.
%H Colin Barker, <a href="/A255236/b255236.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F a(n) = 5*S(n, 6) + S(n-1, 6), n >= 0, with the Chebyshev polynomials S(n, x) (A049310), with S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n-1).
%F G.f.: (5 + x)/(1 - 6*x + x^2).
%F a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(-1) = -1 and a(0) = 5.
%F a(n) = 2*A038761(n) + A038762(n), n >= 0. See the Mar 19 comment on A054490. - _Wolfdieter Lang_, Mar 19 2015
%F a(n) = ((3-2*sqrt(2))^n*(-8+5*sqrt(2)) + (3+2*sqrt(2))^n*(8+5*sqrt(2))) / (2*sqrt(2)). - _Colin Barker_, Oct 13 2015
%e n = 2: 181^2 - 2*(2*64)^2 = -7; (4*64)^2 - 2*181^2 = 14.
%e n = 2: 2*53 + 75 = 181. - _Wolfdieter Lang_, Mar 19 2015
%t CoefficientList[Series[(5 + x) / (1 - 6 x + x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 20 2015 *)
%o (PARI) Vec((5 + x)/(1 - 6*x + x^2) + O(x^30)) \\ _Michel Marcus_, Mar 20 2015
%o (Magma) I:=[5,31]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 20 2015
%Y Cf. A038725, A054490, A038723, A077446, 4*A077447, A254938, A255233.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Feb 26 2015
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