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A090733
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a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.
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4
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2, 25, 623, 15550, 388127, 9687625, 241802498, 6035374825, 150642568127, 3760028828350, 93850078140623, 2342491924687225, 58468448039040002, 1459368709051312825, 36425749278243780623, 909184363247043202750
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OFFSET
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0,1
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COMMENTS
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A Chebyshev T-sequence with Diophantine property.
a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 69*(3*b)^2 =+4 together with the companion sequence b(n)=A097780(n-1), n>=0.
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REFERENCES
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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
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FORMULA
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a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 25)=A097780(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 := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
G.f.: (2-25*x)/(1-25*x+x^2).
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EXAMPLE
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(x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.
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MATHEMATICA
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a[0] = 2; a[1] = 25; a[n_] := 25a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
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PROG
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(Sage) [lucas_number2(n, 25, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 26 2008
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CROSSREFS
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a(n)=sqrt(4 + 69*(3*A097780(n-1))^2), n>=1.
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KEYWORD
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easy,nonn
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AUTHOR
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Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004
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EXTENSIONS
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STATUS
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approved
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