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A077236
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a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.
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8
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4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376
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OFFSET
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0,1
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COMMENTS
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a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).
Bisection (even part) of Chebyshev sequence with Diophantine property.
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LINKS
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FORMULA
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a(n) = T(n+1,2) + 2*T(n,2), with T(n,x) Chebyshev's polynomials of the first kind, A053120. T(n,2) = A001075(n).
G.f.: (4-5*x)/(1-4*x+x^2).
From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start)
a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0.
a(n) = second binomial transform of 4,3,12,9,36. (End)
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EXAMPLE
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11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.
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MATHEMATICA
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CoefficientList[Series[(4-5*x)/(1-4*x+x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{4, -1}, {4, 11}, 30] (* G. C. Greubel, Apr 28 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ G. C. Greubel, Apr 28 2019
(PARI) a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ Michel Marcus, Oct 13 2021
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019
(Sage) ((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019
(GAP) a:=[4, 11];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Apr 28 2019
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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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EXTENSIONS
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STATUS
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approved
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