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A001081
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a(n) = 16*a(n-1) - a(n-2).
(Formerly M4573 N1949)
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15
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1, 8, 127, 2024, 32257, 514088, 8193151, 130576328, 2081028097, 33165873224, 528572943487, 8424001222568, 134255446617601, 2139663144659048, 34100354867927167, 543466014742175624, 8661355881006882817
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OFFSET
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0,2
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COMMENTS
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Chebyshev's polynomials T(n,x) evaluated at x=8.
The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 63*b(n)^2 = +1 with b(n)= A077412(n-1), n>=1 and b(0)=0.
Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(7). - Benoit Cloitre, Feb 14 2004
a(7+14k)-1 and a(7+14k)+1 are consecutive odd powerful numbers. The first pair is 130576328+-1. See A076445. - T. D. Noe, May 04 2006
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REFERENCES
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Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.
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LINKS
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FORMULA
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G.f.: (1-8*x)/(1-16*x+x^2). - Simon Plouffe in his 1992 dissertation.
For all members x of the sequence, 7*x^2 - 7 is a square. Limit_{n->infinity} a(n)/a(n-1) = 8 + 3*sqrt(7). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 8) = (S(n, 16)-S(n-2, 16))/2, with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 16)= A077412(n).
a(n) = ((8 + 3*sqrt(7))^n + (8 - 3*sqrt(7))^n)/2.
a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).
a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - Sture Sjöstedt, Nov 18 2011
a(n) = (-i)^n*Lucas(n, 16*i)/2, where i = sqrt(-1). - G. C. Greubel, Jun 06 2019
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MATHEMATICA
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LinearRecurrence[{16, -1}, {1, 8}, 30]
CoefficientList[Series[(1-8*x)/(1-16*x+x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 20 2017 *)
Table[LucasL[n, 16*I]*(-I)^n/2, {n, 0, 30}] (* G. C. Greubel, Jun 06 2019 *)
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PROG
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(Sage) [lucas_number2(n, 16, 1)/2 for n in range(0, 30)] # Zerinvary Lajos, Jun 26 2008
(Magma) I:=[1, 8]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2013
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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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