A078365 A Chebyshev T-sequence with Diophantine property.

2, 15, 223, 3330, 49727, 742575, 11088898, 165590895, 2472774527, 36926027010, 551417630623, 8234338432335, 122963658854402, 1836220544383695, 27420344506901023, 409468947059131650

Offset

0,1

Comments

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 221*b^2 =+4 with companion sequence b(n)=A078364(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

Harvey P. Dale, Table of n, a(n) for n = 0..851

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 (15,-1).

Formula

a(n)=15*a(n-1)-a(n-2), n >= 1; a(-1)=15, a(0)=2.

a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 15)=A078364(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-15*x)/(1-15*x+x^2).

a(n) = ap^n + am^n, with ap := (15+sqrt(221))/2 and am := (15-sqrt(221))/2.

Mathematica

a[0] = 2; a[1] = 15; a[n_] := 15a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{15, -1}, {2, 15}, 20] (* Harvey P. Dale, Nov 09 2022 *)

Prog

(Sage) [lucas_number2(n, 15, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 26 2008

Crossrefs

a(n)=sqrt(4 + 221*A078364(n-1)^2), n>=1, (Pell equation d=221, +4).

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

Sequence in context: A087962 A140054 A099085 * A207037 A218798 A176337

Adjacent sequences: A078362 A078363 A078364 * A078366 A078367 A078368

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Status

approved