A078367 A Chebyshev T-sequence with Diophantine property.

2, 17, 287, 4862, 82367, 1395377, 23639042, 400468337, 6784322687, 114933017342, 1947076972127, 32985375508817, 558804306677762, 9466687838013137, 160374888939545567, 2716906424134261502, 46027034321342899967, 779742677038695037937, 13209598475336472744962

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 - 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

• Stefano Spezia, Table of n, a(n) for n = 0..800
• 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 (17,-1).

Crossrefs

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

Cf. A049310, A053120, A078366.

Programs

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 *)

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

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

Extensions

More terms from Stefano Spezia, Aug 19 2023