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A005320
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a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.
(Formerly M2919)
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14
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0, 3, 12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308, 12368704813917, 46160634789360, 172273834343523, 642934702584732, 2399464975995405, 8954925201396888, 33420235829592147
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OFFSET
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0,2
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COMMENTS
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For n > 1, a(n-1) is the determinant of the n X n band matrix which has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- and subdiagonal. This matrix appears when constructing a natural cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, Feb 14 2007
The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence whose numerators are the terms of this sequence and denominators are A001075. - Clark Kimberling, Aug 27 2008
a(n) also give the altitude to the middle side of a Super-Heronian Triangle. - Johannes Boot, Oct 14 2010
a(n) gives values of y satisfying 3*x^2 - 4*y^2 = 12; corresponding x values are given by A003500. - Sture Sjöstedt, Dec 19 2017
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REFERENCES
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Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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MAPLE
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a:= n-> (Matrix([[3, 0]]). Matrix([[4, 1], [ -1, 0]])^n)[1, 2]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 14 2008
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MATHEMATICA
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LinearRecurrence[{4, -1}, {0, 3}, 40] (* Harvey P. Dale, Mar 04 2012 *)
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PROG
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(Magma) [3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // G. C. Greubel, Oct 10 2022
(SageMath) [3*chebyshev_U(n-1, 2) for n in range(41)] # G. C. Greubel, Oct 10 2022
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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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