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

0,1

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).

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

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 *)

Prog

(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

Crossrefs

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

Cf. A049310, A053120, A078366.

Sequence in context: A198287 A338635 A268705 * A090306 A304857 A007785

Adjacent sequences: A078364 A078365 A078366 * A078368 A078369 A078370

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Extensions

More terms from Stefano Spezia, Aug 19 2023

Status

approved