A275794 -id:A275794 - cp's OEIS frontend

A276600 Values of m such that m^2 + 6 is a triangular number (A000217).

0, 2, 3, 7, 15, 20, 42, 88, 117, 245, 513, 682, 1428, 2990, 3975, 8323, 17427, 23168, 48510, 101572, 135033, 282737, 592005, 787030, 1647912, 3450458, 4587147, 9604735, 20110743, 26735852, 55980498, 117214000, 155827965, 326278253, 683173257, 908231938

Offset: 1

Colin Barker, Sep 07 2016

2*a(n+2) gives the y members of all positive solutions (x(n), y(n)), proper and improper, of the Pell equation x^2 - 2*y^2 = 7^2, n >= 0. The corresponding x members are x(n) = A106525(n). - Wolfdieter Lang, Sep 29 2016

			7 is in the sequence because 7^2 + 6 = 55, which is a triangular number.

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

Cf. A000217, A230044.
Cf. A001109 (k=0), A106328 (k=1), A077241 (k=2), A276598 (k=3), A276599 (k=5), A276601 (k=9), A276602 (k=10), where k is the value added to n^2.
Cf. A275794, A275796, A106525.

I:=[0,2,3,7,15,20]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Sep 15 2021

LinearRecurrence[{0,0,6,0,0,-1}, {0,2,3,7,15,20}, 41] (* G. C. Greubel, Sep 15 2021 *)

concat(0, Vec(x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) + O(x^40)))

def A276600_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(2+3*x+7*x^2+3*x^3+2*x^4)/(1-6*x^3+x^6) ).list()
a=A276600_list(41); a[1:] # G. C. Greubel, Sep 15 2021

a(n) = 6*a(n-3) - a(n-6) for n>6.
G.f.: x^2*(2 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/(1 - 6*x^3 + x^6).
From Wolfdieter Lang, Sep 29 2016: (Start)
Trisection:
a(2+3*n) = 15*S(n-1,6) - 2*S(n-2,6) = A275794(n),
a(3+3*n) = 20*S(n-1,6) - 3*S(n-2,6) = A275796(n),
a(4+3*n) = 7*(6*S(n-1,6) - S(n-2,6)) = 7*A001109(n+1) for n >= 0, with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
(End)

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

Original entry on oeis.org

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

A275795 The x 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

11, 57, 331, 1929, 11243, 65529, 381931, 2226057, 12974411, 75620409, 440748043, 2568867849, 14972459051, 87265886457, 508622859691, 2964471271689, 17278204770443, 100704757350969, 586950339335371, 3420997278661257, 19939033332632171, 116213202717131769

Offset: 0

Wolfdieter Lang, Sep 27 2016

For details and the Nagell reference see A275793.

The y2(n) members are given in A275795(n).

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.

LinearRecurrence[{6,-1},{11,57},30] (* Harvey P. Dale, Sep 01 2022 *)

a(n) = round((((3-2*sqrt(2))^n*(-12+11*sqrt(2))+(3+2*sqrt(2))^n*(12+11*sqrt(2)))) / (2*sqrt(2))) \\ Colin Barker, Sep 28 2016

Vec((11-9*x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Oct 09 2016

a(n) = 57*S(n-1,6) - 11*S(n-2,6) with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
O.g.f.: (11 - 9*x)/(1 - 6*x + x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = 9 and a(0) = 11.
a(n) = (((3-2*sqrt(2))^n*(-12+11*sqrt(2))+(3+2*sqrt(2))^n*(12+11*sqrt(2)))) / (2*sqrt(2)). - Colin Barker, Sep 28 2016

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

cp's OEIS Frontend

A276600 Values of m such that m^2 + 6 is a triangular number (A000217).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A275795 The x 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula