OFFSET
0,1
COMMENTS
a(n) gives the general (positive integer) solution of the Pell equation a^2 - 357*b^2 =+4 with companion sequence b(n)=A078368(n-1), n>=1.
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)=19*a(n-1)-a(n-2), n >= 1; a(-1)=19, a(0)=2.
a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 19)=A078368(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-19*x)/(1-19*x+x^2).
a(n) = ap^n + am^n, with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.
MATHEMATICA
a[0] = 2; a[1] = 19; a[n_] := 19a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{19, -1}, {2, 19}, 20] (* Harvey P. Dale, Dec 24 2021 *)
PROG
(Sage) [lucas_number2(n, 19, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 27 2008
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
STATUS
approved