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A078363
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A Chebyshev T-sequence with Diophantine property.
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3
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2, 13, 167, 2158, 27887, 360373, 4656962, 60180133, 777684767, 10049721838, 129868699127, 1678243366813, 21687295069442, 280256592535933, 3621648407897687, 46801172710133998, 604793596823844287
(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 - 165*b^2 =+4 with companion sequence b(n)=A078362(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)=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.
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MATHEMATICA
| a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (from Robert G. Wilson v Jan 30 2004)
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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])
sage: [lucas_number2(n, 13, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| a(n)=sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A090643 A132521 A177448 * A143851 A088316 A006905
Adjacent sequences: A078360 A078361 A078362 * A078364 A078365 A078366
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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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