OFFSET
0,1
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
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
EXTENSIONS
More terms from Colin Barker, Feb 26 2014
STATUS
approved