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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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

• 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).

Crossrefs

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

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

Programs

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 *)

Sage

[lucas_number2(n,19,1) for n in range(0,20)] # Zerinvary Lajos, Jun 27 2008

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.