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A078365 A Chebyshev T-sequence with Diophantine property. 3
2, 15, 223, 3330, 49727, 742575, 11088898, 165590895, 2472774527, 36926027010, 551417630623, 8234338432335, 122963658854402, 1836220544383695, 27420344506901023, 409468947059131650 (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 - 221*b^2 =+4 with companion sequence b(n)=A078364(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

Table of n, a(n) for n=0..15.

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 (15,-1).

FORMULA

a(n)=15*a(n-1)-a(n-2), n >= 1; a(-1)=15, a(0)=2.

a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 15)=A078364(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-15*x)/(1-15*x+x^2).

a(n) = ap^n + am^n, with ap := (15+sqrt(221))/2 and am := (15-sqrt(221))/2.

a(n)=(15/2)*[15/2+(1/2)*sqrt(221)]^n-(1/2)*[15/2+(1/2)*sqrt(221)]^n*sqrt(221)+(1/2)*sqrt(221) *[15/2-(1/2)*sqrt(221)]^n+(15/2)*[15/2-(1/2)*sqrt(221)]^n, with n>=0 - Paolo P. Lava, Jun 19 2008

MATHEMATICA

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

PROG

(Sage) [lucas_number2(n, 15, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 26 2008

CROSSREFS

a(n)=sqrt(4 + 221*A078364(n-1)^2), n>=1, (Pell equation d=221, +4).

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

Sequence in context: A087962 A140054 A099085 * A207037 A218798 A176337

Adjacent sequences:  A078362 A078363 A078364 * A078366 A078367 A078368

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 29 2002

STATUS

approved

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Last modified November 18 09:05 EST 2017. Contains 294865 sequences.