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A078365
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A Chebyshev T-sequence with Diophantine property.
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3
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2, 15, 223, 3330, 49727, 742575, 11088898, 165590895, 2472774527, 36926027010, 551417630623, 8234338432335, 122963658854402, 1836220544383695, 27420344506901023, 409468947059131650
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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.
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REFERENCES
| O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
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.
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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 (paoloplava(AT)gmail.com), Jun 19 2008
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MATHEMATICA
| a[0] = 2; a[1] = 15; a[n_] := 15a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (from Robert G. Wilson v Jan 30 2004)
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PROG
| sage: [lucas_number2(n, 15, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
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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 * A176337 A145168 A184357
Adjacent sequences: A078362 A078363 A078364 * A078366 A078367 A078368
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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