A078363 A Chebyshev T-sequence with Diophantine property.

2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242

Offset: 0

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.

Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014

References

• O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (13,-1).

Crossrefs

Cf. A078362.

Cf. A077428, A078355 (Pell +4 equations).

Cf. A049310, A053120.

Programs

Mathematica

a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{13,-1},{2,13},20] (* Harvey P. Dale, Oct 28 2016 *)

PARI

a(n)=if(n<0,0,2*subst(poltchebi(n),x,13/2))

PARI

a(n)=if(n<0,0,polsym(1-13*x+x^2,n)[n+1])

PARI

Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014

Sage

[lucas_number2(n,13,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.

a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-13*x)/(1-13*x+x^2).

a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.

a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).

E.g.f.: 2*exp(13*x/2)*cosh(sqrt(165)*x/2). - Stefano Spezia, Sep 24 2022

Extensions

More terms from Colin Barker, Feb 26 2014