A275795 -id:A275795 - cp's OEIS frontend

A106525 Values of x in x^2 - 49 = 2*y^2.

9, 11, 21, 43, 57, 119, 249, 331, 693, 1451, 1929, 4039, 8457, 11243, 23541, 49291, 65529, 137207, 287289, 381931, 799701, 1674443, 2226057, 4660999, 9759369, 12974411, 27166293, 56881771, 75620409, 158336759, 331531257, 440748043

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), May 07 2005

From Wolfdieter Lang, Sep 27 2016: (Start)
These are the x members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2.
The y(n) members are given in 2*A276600(n+2).
This sequence is composed of the two y members of the two proper classes of solutions of the Pell equation x^2 - 2*y^2 = 7^2 and of 7 times the proper solutions X of the Pell equation X^2 - 2*Y^2 = +1. See A275793, A275795 and 7*A001541. See A275793 for further information, and the Nagell reference. (End)
The sums of the consecutive integers in the following sequences will be squares: for n, i >= 1, if mod(i,3)=0 then 7*n+1, 7*n+2, ..., a(i)*n + (A001541(i/3)-1)/2; otherwise, if mod(i,3)=1 or 2 then 7*n+4, 7*n+5, ..., a(i)*n + (a(i)-1)/2.

			In the following, aa(n) denotes A001541(n):
a(9)=693; as mod(9,3)=0, a(9)=aa(3)*7=99*7=693, also 693^2-49=2*490^2
a(10)=1451; as mod(10,3)=1, a(10)=(aa(5)+aa(2)+aa(3)-aa(4))/2 =(3363+17+99-577)/2=1451, also 1451^2-49=2*1026^2.
The solutions (proper and every third pair improper) of x^2 - 2*y^2 = +49 begin [9, 4], [11, 6], [21, 14], [43, 30], [57, 40], [119, 84], [249, 176], [331, 234], [693, 490], [1451, 1026], [1929, 1364], [4039, 2856], [8457, 5980], [11243, 7950], [23541, 16646], ... - _Wolfdieter Lang_, Sep 27 2016

Colin Barker, Table of n, a(n) for n = 1..1001 (offset adapted by _Georg Fischer_, Jan 31 2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. 14*A001109, 7*A001541, A106525, A106526.

Cf. A275793, A275794, A275795, A275796, 2*A276600.

I:=[9,11,21,43,57,119]; [n le 6 select I[n] else 6*Self(n-3)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 26 2018

LinearRecurrence[{0,0,6,0,0,-1}, {9,11,21,43,57,119}, 50] (* Vincenzo Librandi, Oct 26 2018 *)

Vec((9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) + O(x^50)) \\ Colin Barker, Sep 28 2016

def A106525_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(9+11*x+21*x^2-11*x^3-9*x^4-7*x^5)/(1-6*x^3+x^6) ).list()
a=A106525_list(41); a[1:] # G. C. Greubel, Sep 15 2021

a(3*k)   = A001541(k)*7 for k >= 2.
a(3*k+1) = (A001541(k+2) + A001541(k-1) + A001541(k) - A001541(k+1))/2;
a(3*k+2) = (A001541(k+2) + A001541(k-1) - A001541(k) + A001541(k+1))/2.
a(3*n) = A275793(n), a(3*n+1) = A275795(n), a(3*n+2) = 7*A001541(n+1), n >= 0. - Wolfdieter Lang, Sep 27 2016
From Colin Barker, Mar 29 2012: (Start)
a(n) = 6*a(n-3) - a(n-6).
G.f.: x*(9 + 11*x + 21*x^2 - 11*x^3 - 9*x^4 - 7*x^5)/(1 - 6*x^3 + x^6). (End)

Entry revised by N. J. A. Sloane, Oct 26 2018 at the suggestion of Georg Fischer.

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.

Original entry on oeis.org

9, 43, 249, 1451, 8457, 49291, 287289, 1674443, 9759369, 56881771, 331531257, 1932305771, 11262303369, 65641514443, 382586783289, 2229879185291, 12996688328457, 75750250785451, 441504816384249, 2573278647520043, 14998167068736009, 87415723764896011

Offset: 0

Wolfdieter Lang, Sep 27 2016

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)).

			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], ...

T. Nagell, Introduction to Number Theory, Wiley, 1951, Theorem 109, pp. 207-208.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A001109, A058529, A106525.

Cf. A263012, A275794, A275795, A275796, 2*A276600.

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

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 *)

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

Vec((9-11*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 02 2016

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

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

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.

Original entry on oeis.org

3, 20, 117, 682, 3975, 23168, 135033, 787030, 4587147, 26735852, 155827965, 908231938, 5293563663, 30853150040, 179825336577, 1048098869422, 6108767879955, 35604508410308, 207518282581893, 1209505187081050, 7049512839904407

Offset: 0

Wolfdieter Lang, Sep 27 2016

For the x2(n) members see A275795(n).

For details and the Nagell reference see A275793.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A275793, A275794, A275795.

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

Vec((3 + 2*x)/(1 - 6*x + x^2) + O(x^20)) \\ Felix Fröhlich, Sep 28 2016

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

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A106525 Values of x in x^2 - 49 = 2*y^2.

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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.

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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.

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