A041758 -id:A041758 - cp's OEIS frontend

A041759 Denominators of continued fraction convergents to sqrt(399).

1, 1, 39, 40, 1559, 1599, 62321, 63920, 2491281, 2555201, 99588919, 102144120, 3981065479, 4083209599, 159143030241, 163226239840, 6361740144161, 6524966384001, 254310462736199, 260835429120200, 10166056769303799, 10426892198423999

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 38 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,40,0,-1).

Cf. A041758, A040379, A002530.

I:=[1,1,39,40]; [n le 4 select I[n] else 40*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2013

Denominator[Convergents[Sqrt[399], 30]] (* Vincenzo Librandi, Dec 24 2013 *)

G.f.: -(x^2-x-1) / (x^4-40*x^2+1). - Colin Barker, Nov 24 2013
a(n) = 40*a(n-2) - a(n-4) for n > 3. - Vincenzo Librandi, Dec 24 2013
From Peter Bala, May 27 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(38) + sqrt(42) )/2 and beta = ( sqrt(38) - sqrt(42) )/2 be the roots of the equation x^2 - sqrt(38)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 38 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 38*a(2*n) + a(2*n - 1). (End)

More terms from Colin Barker, Nov 24 2013

A322890 a(n) = value of Chebyshev T-polynomial T_n(20).

Original entry on oeis.org

1, 20, 799, 31940, 1276801, 51040100, 2040327199, 81562047860, 3260441587201, 130336101440180, 5210183616019999, 208277008539359780, 8325870157958371201, 332826529309795488260, 13304735302233861159199, 531856585560044650879700

Offset: 0

Seiichi Manyama, Dec 29 2018

Seiichi Manyama, Table of n, a(n) for n = 0..624
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (40, -1).

Column 20 of A322836.

Cf. A041758.

a:=[1,20];; for n in [3..20] do a[n]:=40*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 31 2018

seq(coeff(series((1-20*x)/(1-40*x+x^2),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Dec 31 2018

CoefficientList[Series[(1 - 20 x)/(1 - 40 x + x^2), {x, 0, 15}], x] (* or *)
Array[ChebyshevT[#, 20] &, 16, 0] (* Michael De Vlieger, Jan 01 2019 *)

PARI
```
{a(n) = polchebyshev(n, 1, 20)}
```

Vec((1 - 20*x) / (1 - 40*x + x^2) + O(x^20)) \\ Colin Barker, Dec 30 2018

a(0) = 1, a(1) = 20 and a(n) = 40*a(n-1) - a(n-2) for n > 1.
From Colin Barker, Dec 30 2018: (Start)
G.f.: (1 - 20*x) / (1 - 40*x + x^2).
a(n) = ((20+sqrt(399))^(-n) * (1+(20+sqrt(399))^(2*n))) / 2.
(End)

cp's OEIS Frontend

A041759 Denominators of continued fraction convergents to sqrt(399).

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