OFFSET
0,1
COMMENTS
a(n) gives the general (positive integer) solution of the Pell equation a^2 - 285*b^2 =+4 with companion sequence b(n)=A078366(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) = 17*a(n-1)-a(n-2), n >= 1; a(-1)=17, a(0)=2.
a(n) = sqrt(4 + 285*A078366(n-1)^2), n>=1, (Pell equation d=285, +4).
a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 17)=A078366(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2-17*x)/(1-17*x+x^2).
a(n) = ap^n + am^n, with ap := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.
E.g.f.: 2*exp(17*x/2)*cosh(sqrt(285)*x/2). - Stefano Spezia, Aug 19 2023
MATHEMATICA
a[0] = 2; a[1] = 17; a[n_] := 17a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
PROG
(PARI) a(n)=if(n<0, 0, subst(2*poltchebi(n), x, 17/2))
(Sage) [lucas_number2(n, 17, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 26 2008
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
EXTENSIONS
More terms from Stefano Spezia, Aug 19 2023
STATUS
approved