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A077244
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Bisection (odd part) of Chebyshev sequence with Diophantine property.
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5
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3, 22, 173, 1362, 10723, 84422, 664653, 5232802, 41197763, 324349302, 2553596653, 20104423922, 158281794723, 1246149933862, 9810917676173, 77241191475522, 608118614128003, 4787707721548502, 37693543158260013, 296760637544531602, 2336391557197992803
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OFFSET
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0,1
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COMMENTS
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3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077243(n).
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LINKS
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FORMULA
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a(n)= (2*T(n+1, 4)+T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).
G.f.: (3-2*x)/(1-8*x+x^2).
a(n) = (((4-sqrt(15))^n * (-10+3*sqrt(15)) + (4+sqrt(15))^n * (10+3*sqrt(15)))) / (2*sqrt(15)).
a(n) = 8*a(n-1) - a(n-2).
(End)
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EXAMPLE
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22 = a(1) = sqrt((5*A077243(1)^2 + 7)/3) = sqrt((5*17^2 + 7)/3) = sqrt(484) = 22.
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MATHEMATICA
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PROG
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(PARI) Vec((3-2*x)/(1-8*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015
(Magma) I:=[3, 22]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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