A090729 a(n) = 21a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 21.

2, 21, 439, 9198, 192719, 4037901, 84603202, 1772629341, 37140612959, 778180242798, 16304644485799, 341619353958981, 7157701788652802, 149970118207749861, 3142214780574094279, 65836540273848229998

Offset

0,1

Comments

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 437*b^2 =+4 with companion sequence b(n)=A092499(n), n>=0.

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

Indranil Ghosh, Table of n, a(n) for n = 0..755

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 (21,-1).

Formula

a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 21)=A092499(n+1). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.

G.f.: (2-21*x)/(1-21*x+x^2).

Mathematica

a[0] = 2; a[1] = 21; a[n_] := 21a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

Prog

(Sage) [lucas_number2(n, 21, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 27 2008

Crossrefs

Cf. A085985.

a(n)=sqrt(4 + 437*A092499(n)^2), n>=1, (Pell equation d=437, +4).

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

Sequence in context: A091315 A359716 A087546 * A090310 A024232 A192666

Adjacent sequences: A090726 A090727 A090728 * A090730 A090731 A090732

Keyword

easy,nonn

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

Extensions

Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004

Status

approved