A078369 A Chebyshev T-sequence with Diophantine property.

2, 19, 359, 6802, 128879, 2441899, 46267202, 876634939, 16609796639, 314709501202, 5962870726199, 112979834296579, 2140653980908802, 40559445802970659, 768488816275533719, 14560728063432170002

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

Table of n, a(n) for n=0..15.

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 (19,-1).

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

a(n)=sqrt(4 + 357*A078368(n-1)^2), n>=1, (Pell equation d=357, +4).

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

Sequence in context: A233107 A187659 A308330 * A090308 A110818 A325288

Adjacent sequences: A078366 A078367 A078368 * A078370 A078371 A078372

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved