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%I M1540 #90 Oct 25 2022 16:58:24
%S 2,5,23,110,527,2525,12098,57965,277727,1330670,6375623,30547445,
%T 146361602,701260565,3359941223,16098445550,77132286527,369562987085,
%U 1770682648898,8483850257405,40648568638127,194758992933230,933146396028023,4470972987206885
%N a(n) = 5*a(n-1) - a(n-2), with a(0) = 2, a(1) = 5.
%C Positive values of x satisfying x^2 - 21*y^2 = 4; values of y are in A004254. - _Wolfdieter Lang_, Nov 29 2002
%C Except for the first term, positive values of x (or y) satisfying x^2 - 5xy + y^2 + 21 = 0. - _Colin Barker_, Feb 08 2014
%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="/A003501/b003501.txt">Table of n, a(n) for n = 0..200</a>
%H Peter Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a>
%H Noureddine Chair, <a href="https://doi.org/10.1016/j.aop.2012.09.002">Exact two-point resistance, and the simple random walk on the complete graph minus N edges</a>, Ann. Phys. 327, No. 12, 3116-3129 (2012), P(7).
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Lisa Lokteva, <a href="https://arxiv.org/abs/2208.14850">Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls</a>, arXiv:2208.14850 [math.GT], 2022.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Jeffrey Shallit, <a href="http://www.jstor.org/stable/2690344">An interesting continued fraction</a>, Math. Mag., 48 (1975), 207-211.
%H Jeffrey Shallit, <a href="/A005248/a005248_1.pdf">An interesting continued fraction</a>, Math. Mag., 48 (1975), 207-211. [Annotated scanned copy]
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2), with S(n, x)=U(n, x/2), S(-1, x)=0, S(-2, x)=-1. U(n, x), resp. T(n, x), are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 5) = A004254(n), n>=0.
%F G.f.: (2-5*x)/(1-5*x+x^2). - _Simon Plouffe_ in his 1992 dissertation.
%F a(n) ~ (1/2*(5 + sqrt(21)))^n. - Joe Keane (jgk(AT)jgk.org), May 16 2002
%F a(n) = ap^n + am^n, with ap=(5+sqrt(21))/2 and am=(5-sqrt(21))/2.
%F a(n) = sqrt(4 + 21*A004254(n)^2).
%F From _Peter Bala_, Jan 06 2013: (Start)
%F Let F(x) = Product_{n=0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(5 - sqrt(21)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.19827 65373 95327 17782 ... = 2 + 1/(5 + 1/(23 + 1/(110 + ...))).
%F Also F(-alpha) = 0.79824 49142 28050 93561 ... has the continued fraction representation 1 - 1/(5 - 1/(23 - 1/(110 - ...))) and the simple continued fraction expansion 1/(1 + 1/((5-2) + 1/(1 + 1/((23-2) + 1/(1 + 1/((110-2) + 1/(1 + ...))))))).
%F F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((5^2-4) + 1/(1 + 1/((23^2-4) + 1/(1 + 1/((110^2-4) + 1/(1 + ...))))))).
%F (End)
%F a(n) = (A217787(k+3n) + A217787(k-3n))/A217787(k) for k>=3n. - _Bruno Berselli_, Mar 25 2013
%e G.f. = 2 + 5*x + 23*x^2 + 110*x^3 + 527*x^4 + 2525*x^5 + ... - _Michael Somos_, Oct 25 2022
%p seq( simplify(2*ChebyshevT(n, 5/2)), n=0..30); # _G. C. Greubel_, Jan 16 2020
%t a[0]=2; a[1]=5; a[n_]:= 5a[n-1] -a[n-2]; Table[a[n], {n,0,30}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t LinearRecurrence[{5,-1},{2,5},30] (* _Harvey P. Dale_, May 12 2019 *)
%t 2*ChebyshevT[Range[0, 30], 5/2] (* _G. C. Greubel_, Jan 16 2020 *)
%t a[ n_] := LucasL[n, 5*I]/I^n; (* _Michael Somos_, Oct 25 2022 *)
%o (PARI) {a(n) = subst(poltchebi(n),x,5/2)*2};
%o (PARI) {a(n) = polchebyshev(n,1,5/2)*2 }; /* _Michael Somos_, Oct 25 2022 */
%o (Sage) [lucas_number2(n,5,1) for n in range(37)] # _Zerinvary Lajos_, Jun 25 2008
%o (Magma) I:=[2,5]; [n le 2 select I[n] else 5*Self(n-1) -Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 16 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 25); Coefficients(R!((2-5*x)/(1-5*x+x^2))); // _Marius A. Burtea_, Jan 16 2020
%o (GAP) a:=[2,5];; for n in [4..30] do a[n]:=5*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 16 2020
%Y Cf. A004252, A004253, A217787.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E Chebyshev comments from _Wolfdieter Lang_, Oct 31 2002
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