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A077409
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Bisection (even part) of Chebyshev sequence with Diophantine property.
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6
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7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811
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OFFSET
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0,1
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COMMENTS
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a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).
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LINKS
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FORMULA
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a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.
a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).
G.f.: (7-11*x)/(1-10*x+x^2).
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EXAMPLE
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59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.
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MATHEMATICA
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LinearRecurrence[{10, -1}, {7, 59}, 30] (* G. C. Greubel, Jan 18 2018 *)
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PROG
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(PARI) a(n)=if(n<0, 0, subst(poltchebi(n+1)+2*poltchebi(n), x, 5))
(PARI) Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015
(Magma) I:=[7, 59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
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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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