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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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).

Crossrefs

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

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

Programs

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 *)

Sage

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

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.