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A078367
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A Chebyshev T-sequence with Diophantine property.
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3
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2, 17, 287, 4862, 82367, 1395377, 23639042, 400468337, 6784322687, 114933017342, 1947076972127, 32985375508817, 558804306677762, 9466687838013137, 160374888939545567, 2716906424134261502
(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 - 285*b^2 =+4 with companion sequence b(n)=A078366(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)=17*a(n-1)-a(n-2), n >= 1; a(-1)=17, a(0)=2.
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.
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MATHEMATICA
| a[0] = 2; a[1] = 17; a[n_] := 17a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (from Robert G. Wilson v Jan 30 2004)
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PROG
| (PARI) a(n)=if(n<0, 0, subst(2*poltchebi(n), x, 17/2))
sage: [lucas_number2(n, 17, 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 + 285*A078366(n-1)^2), n>=1, (Pell equation d=285, +4).
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A176585 A086534 A198287 * A090306 A007785 A201785
Adjacent sequences: A078364 A078365 A078366 * A078368 A078369 A078370
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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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