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A275793
The x members of the positive proper solutions (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - 2*y^2 = +7^2.
5
9, 43, 249, 1451, 8457, 49291, 287289, 1674443, 9759369, 56881771, 331531257, 1932305771, 11262303369, 65641514443, 382586783289, 2229879185291, 12996688328457, 75750250785451, 441504816384249, 2573278647520043, 14998167068736009, 87415723764896011
OFFSET
0,1
COMMENTS
This gives the (increasingly sorted) positive x members of the first class of the proper solutions (x1(n), y1(n)) to the Pell equation x^2 - 2*y^2 = +7^2. For the y1(n) solutions see 2*A275794(n). The solutions for the second class (x2(n), y2(n)) are found in A275795(n) and 2*A275796(n).
All solutions, including the improper ones, are given in A106525(n) and 2*A276600(n+2).
See also the comments on A263012 which apply here mutatis mutandis.
This is for the Pell equations x^2 - 2*y^2 = z^2, besides z^2 = 1 the first instance with proper solutions. For z^2 > 1 there seem to be always two classes of such solutions. For z^2 = 1 there is only one class of proper solutions. These z^2 values seem to appear for z from A058529 (prime factors are +1 or -1 (mod 8)).
REFERENCES
T. Nagell, Introduction to Number Theory, Wiley, 1951, Theorem 109, pp. 207-208.
FORMULA
a(n) = 43*S(n-1, 6) - 9*S(n-2, 6), with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
O.g.f: (9 - 11*x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2) for n >= 1, with a(-1) = 11 and a(0) = 9.
a(n) = (((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2)). - Colin Barker, Sep 28 2016
a(n) = 9*A000129(2*n+1) - A000129(2*n). - G. C. Greubel, Sep 15 2021
EXAMPLE
The first positive proper fundamental solution (x = x1(n), y = y1(n)) of x^2 - 2*y^2 = 49 are [9, 4], [43, 30], [249, 176], [1451, 1026], [8457, 5980], [49291, 34854], [287289, 203144], [1674443, 1184010], ...
The first positive proper fundamental solution of the second class (x = x2(n), y = y2(n)) are [11, 6], [57, 40], [331, 234], [1929, 1364], [11243, 7950], [65529, 46336], [381931, 270066], [2226057, 1574060], ...
MATHEMATICA
RecurrenceTable[{a[n]== 6a[n-1] -a[n-2], a[-1]==11, a[0]==9}, a, {n, 0, 25}] (* Michael De Vlieger, Sep 28 2016 *)
Table[9*Fibonacci[2*n+1, 2] - Fibonacci[2*n, 2], {n, 0, 30}] (* G. C. Greubel, Sep 15 2021 *)
PROG
(PARI) a(n) = round((((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2))) \\ Colin Barker, Sep 28 2016
(PARI) Vec((9-11*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 02 2016
(Magma) I:=[9, 43]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..31]]; // G. C. Greubel, Sep 15 2021
(Sage)
def P(n): return lucas_number1(n, 2, -1);
[9*P(2*n+1) - P(2*n) for n in (0..30)] # G. C. Greubel, Sep 15 2021
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 27 2016
STATUS
approved