A188645 -id:A188645 - cp's OEIS frontend

A099397 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=51.

1, 51, 5201, 530451, 54100801, 5517751251, 562756526801, 57395647982451, 5853793337683201, 597029524795704051, 60891157735824130001, 6210301059529265556051, 633389816914249262587201

Offset: 0

Wolfdieter Lang, Oct 18 2004

Used in A099374.

Numbers n such that 26*(n^2-1) is square. - Vincenzo Librandi, Nov 17 2010

Indranil Ghosh, Table of n, a(n) for n = 0..497
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (102, -1).

Row 5 of array A188645.

[n: n in [1..1000] |IsSquare(26*(n^2-1))] // Vincenzo Librandi, Nov 17 2010

LinearRecurrence[{102, -1},{1, 51},13] (* Ray Chandler, Aug 11 2015 *)

a(n) = polchebyshev(n, 1, 51); \\ Michel Marcus, Jan 20 2018

a(n) = 102*a(n-1) - a(n-2), n>=1; a(-1):= 51, a(0)=1.
a(n) = T(n, 51) = (S(n, 102)-S(n-2, 102))/2 = S(n, 102)-51*S(n-1, 102) with T(n, x), resp. S(n, x), Chebyshev polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 102)=(n).
a(n) = (ap^n + am^n)/2 with ap := 51+10*sqrt(26) and am := 51-10*sqrt(26).
a(n) = Sum_{k=0..floor(n/2)} (((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*51)^(n-2*k)), n >= 1. a(0):=1.
G.f.: (1 - 51*x)/(1 - 102*x + x^2).

A174747 x-values in the solution to x^2-37*y^2=1.

Original entry on oeis.org

1, 73, 10657, 1555849, 227143297, 33161365513, 4841332221601, 706801342988233, 103188154744060417, 15064763791289832649, 2199352325373571506337, 321090374740750150092553, 46876995359824148342006401

Offset: 1

Vincenzo Librandi, Apr 12 2010

The corresponding values of y of this Pell equation are in A174775

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (146,-1).

Cf. A174775, Row 6 of array A188645.

I:=[1, 73]; [n le 2 select I[n] else 146*Self(n-1)-Self(n-2): n in [1..20]];

Mathematica
```
LinearRecurrence[{146,-1},{1,73},30]
```

a(n) = 146*a(n-1)-a(n-2) with a(1)=1, a(2)=73.

G.f.: x*(1-73*x)/(1-146*x+x^2). - Bruno Berselli, Apr 18 2011

A176368 x-values in the solution to x^2 - 65*y^2 = 1.

Original entry on oeis.org

1, 129, 33281, 8586369, 2215249921, 571525893249, 147451465208321, 38041906497853569, 9814664424981012481, 2532145379738603366529, 653283693308134687552001, 168544660728119010785049729

Offset: 1

Vincenzo Librandi, Apr 16 2010

The corresponding values of y of this Pell equation are in A176369.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (258,-1).

Cf. A176369, Row 8 of array A188645.

a:=[1,129];; for n in [3..15] do a[n]:=258*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 08 2019

I:=[1, 129]; [n le 2 select I[n] else 258*Self(n-1)-Self(n-2): n in [1..20]];

seq(coeff(series(x*(1-129*x)/(1-258*x+x^2), x, n+1), x, n), n = 1..15); # G. C. Greubel, Dec 08 2019

Mathematica
```
LinearRecurrence[{258,-1},{1,129},30]
```

my(x='x+O('x^15)); Vec(x*(1-129*x)/(1-258*x+x^2)) \\ G. C. Greubel, Dec 08 2019

def A176368_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-129*x)/(1-258*x+x^2) ).list()
a=A176368_list(15); a[1:] # G. C. Greubel, Dec 08 2019

a(n) = 258*a(n-1) - a(n-2) with a(1)=1, a(2)=129.

G.f.: x*(1-129*x)/(1-258*x+x^2).

cp's OEIS Frontend

A099397 Chebyshev polynomial of the first kind, T(n,x), evaluated at x=51.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A174747 x-values in the solution to x^2-37*y^2=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A176368 x-values in the solution to x^2 - 65*y^2 = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula