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URL: https://oeis.org/A275796

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A275796
One half of the y members of the positive proper solutions (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - 2*y^2 = +7^2.
4
3, 20, 117, 682, 3975, 23168, 135033, 787030, 4587147, 26735852, 155827965, 908231938, 5293563663, 30853150040, 179825336577, 1048098869422, 6108767879955, 35604508410308, 207518282581893, 1209505187081050, 7049512839904407, 41087571852345392, 239475918274167945, 1395767937792662278
OFFSET
0,1
COMMENTS
For the x2(n) members see A275795(n).
For details and the Nagell reference see A275793.
FORMULA
a(n) = 20*S(n-1,6) - 3*S(n-2,6), with the Chebyshev polynomials S(n, 6) = A001109(n+1) for n >= -1, with S(-2, 6) = -1.
O.g.f: (3 + 2*x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2) for n >= 1, with a(-1) = -2 and a(0) = 3.
a(n) = (((3-2*sqrt(2))^n*(-11+6*sqrt(2))+(3+2*sqrt(2))^n*(11+6*sqrt(2)))) / (4*sqrt(2)). - Colin Barker, Sep 28 2016
MATHEMATICA
LinearRecurrence[{6, -1}, {3, 20}, 25] (* Paolo Xausa, Jan 11 2026 *)
PROG
(PARI) a(n) = round((((3-2*sqrt(2))^n*(-11+6*sqrt(2))+(3+2*sqrt(2))^n*(11+6*sqrt(2))))/(4*sqrt(2))) \\ Colin Barker, Sep 28 2016
(PARI) Vec((3 + 2*x)/(1 - 6*x + x^2) + O(x^20)) \\ Felix Fröhlich, Sep 28 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 27 2016
STATUS
approved