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A090248
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a(n) =27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27.
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4
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2, 27, 727, 19602, 528527, 14250627, 384238402, 10360186227, 279340789727, 7531841136402, 203080369893127, 5475638145978027, 147639149571513602, 3980781400284889227, 107333458658120495527, 2894022602368968490002
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n+1)/a(n) converges to ((27+sqrt(725))/2) =26.96291201... Lim a(n)/a(n+1) as n approaches infinity = 0.03708798... =2/(27+sqrt(725)) =(27-sqrt(725))/2. Lim a(n+1)/a(n) as n approaches infinity = 26.96291201... = (27+sqrt(725))/2=2/(27-sqrt(725)). Lim a(n)/a(n+1) = 27 - Lim a(n+1)/a(n).
A Chebyshev T-sequence with Diophantine property.
a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 29*(5*b)^2 =+4 with companion sequence b(n)=A097781(n-1), n>=0.
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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
| 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) =27a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 27. a(n) = ((27+sqrt(725))/2)^n + ((27-sqrt(725))/2)^n, (a(n))^2 =a(2n)+2.
a(n) = S(n, 27) - S(n-2, 27) = 2*T(n, 27/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (27+5*sqrt(29))/2 and am := (27-5*sqrt(29))/2.
G.f.: (2-27*x)/(1-27*x+x^2).
a(-n) = a(n). - Michael Somos Nov 01 2008
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EXAMPLE
| a(4) =528527 = 27a(3) - a(2) = 27*19602 - 727= ((27+sqrt(725))/2)^4 + ((27-sqrt(725))/2)^4 = 528526.999998107 + 0.000001892 =528527.
(x;y) = (2;0), (27;1), (727;27), (19602;728), ... give the nonnegative integer solutions to x^2 - 29*(5*y)^2 =+4.
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MATHEMATICA
| a[0] = 2; a[1] = 27; a[n_] := 27a[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, 27, 1) for n in xrange(0, 16)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008
(PARI) {a(n) = (-1)^n * subst(2 * poltchebi(2*n), 'x, -5/2 * I)} /* Michael Somos Nov 04 2008 */
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CROSSREFS
| Cf. A046213, A078046.
a(n)=sqrt(4 + 29*(5*A097781(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A090733 for 2*T(n, 25/2).
A087130(2*n) = a(n). - Michael Somos Nov 01 2008
Sequence in context: A203429 A153850 A138458 * A182934 A078102 A067075
Adjacent sequences: A090245 A090246 A090247 * A090249 A090250 A090251
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KEYWORD
| easy,nonn
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AUTHOR
| Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 30 2004
Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004
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