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A078363 A Chebyshev T-sequence with Diophantine property. 3
2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287, 7815515585999841733, 100996909021174098242 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 = +4 with companion sequence b(n)=A078362(n-1), n>=1.

Except for the first term, positive values of x (or y) satisfying x^2 - 13xy + y^2 + 165 = 0. - Colin Barker, Feb 26 2014

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

Table of n, a(n) for n=0..18.

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 (13,-1).

FORMULA

a(n) = 13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.

a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-13*x)/(1-13*x+x^2).

a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.

a(n) = sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).

MATHEMATICA

a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)

LinearRecurrence[{13, -1}, {2, 13}, 20] (* Harvey P. Dale, Oct 28 2016 *)

PROG

(PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 13/2))

(PARI) a(n)=if(n<0, 0, polsym(1-13*x+x^2, n)[n+1])

(PARI) Vec((2-13*x)/(1-13*x+x^2) + O(x^100)) \\ Colin Barker, Feb 26 2014

(Sage) [lucas_number2(n, 13, 1) for n in xrange(0, 20)] - Zerinvary Lajos, Jun 25 2008

CROSSREFS

Cf. A078362.

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

Sequence in context: A132521 A177448 A258224 * A143851 A088316 A006905

Adjacent sequences:  A078360 A078361 A078362 * A078364 A078365 A078366

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

EXTENSIONS

More terms from Colin Barker, Feb 26 2014

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.