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%I M3637 N1479
%S 1,4,31,244,1921,15124,119071,937444,7380481,58106404,457470751,
%T 3601659604,28355806081,223244789044,1757602506271,13837575261124,
%U 108942999582721,857706421400644,6752708371622431,53163960551578804
%N a(n) = 8a(n-1) - a(n-2); a(0) = 1, a(1) = 4.
%C a(15+30k)-1 and a(15+30k)+1 are consecutive odd powerful numbers. The first pair is 13837575261124+-1. See A076445. - _T. D. Noe_, May 04 2006
%C Numbers n such that 15*(n^2-1) is a square. [From _Vincenzo Librandi_, Jul 08 2010]
%C a(n) solves for x in the Diophantine equation x^2-15*y^2=1, The corresponding y solutions are provided by A001090. [From _Vincenzo Librandi_
%D 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
%D H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001091/b001091.txt">Table of n, a(n) for n=0..200</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H _Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>
%F For all elements x of the sequence, 15*x^2 -15 is a square. Lim. n -> Inf. a(n)/a(n-1) = 4 + sqrt(15). - _Gregory V. Richardson_, Oct 11 2002
%F a(n) = ((4+sqrt(15))^n + (4-sqrt(15))^n)/2.
%F a(n) = 4*S(n-1, 8)-S(n-2, 8) = (S(n, 8)-S(n-2, 8))/2, n>=1; S(n, x) := U(n, x/2) with Chebyshev's polynomials of the 2nd kind, A049310, with S(-1, x) := 0 and S(-2, x) := -1.
%F a(n) = T(n, 4) with Chebyshev's polynomials of the first kind; see A053120.
%F G.f.: (1-4*x)/(1-8*x+x^2). a(n)=a(-n). - R. Stephan, Jun 06 2005
%F a(n)a(n+3) - a(n+1)a(n+2) = 120. - R. Stephan, Jun 06 2005
%p A001091:=-(-1+4*z)/(1-8*z+z**2); [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%o (PARI) a(n)=subst(poltchebi(n),x,4)
%o (PARI) a(n)=n=abs(n); polcoeff((1-4*x)/(1-8*x+x^2)+x*O(x^n),n) /* Michael Somos Jun 07 2005 */
%Y a(n) = sqrt{15*[(A001090(n))^2]+1}.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_.
%E More terms from Larry Reeves (larryr(AT)acm.org), Aug 25 2000
%E Chebyshev comments from _Wolfdieter Lang_, Oct 31 2002
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