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%I #69 Sep 08 2022 08:45:11
%S 2,10,98,970,9602,95050,940898,9313930,92198402,912670090,9034502498,
%T 89432354890,885289046402,8763458109130,86749292044898,
%U 858729462339850,8500545331353602,84146723851196170,832966693180608098,8245520207954884810
%N a(n) = 10*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10.
%C a(n+1)/a(n) converges to (5+sqrt(24)) = 9.8989794... a(0)/a(1)=2/10; a(1)/a(2)=10/98; a(2)/a(3)=98/970; a(3)/a(4)=970/9602; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.10102051... = 1/(5+sqrt(24)) = (5-sqrt(24)).
%C Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 96 = 0. - _Colin Barker_, Feb 25 2014
%C A triangle whose sides are a(n) - 1, a(n) and a(n) + 1 is nearly Fleenor-Heronian since its area is the product of an integer and the square root of 2. See A003500. - _Charlie Marion_, Dec 18 2020
%H T. D. Noe, <a href="/A087799/b087799.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 Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%F a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n.
%F G.f.: (2-10*x)/(1-10*x+x^2). - _Philippe Deléham_, Nov 02 2008
%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 = 5 - sqrt(24). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.09989 80642 72052 68138 ... = 2 + 1/(10 + 1/(98 + 1/(970 + ...))).
%F Also F(-alpha) = 0.89989 78538 78393 34715 ... has the continued fraction representation 1 - 1/(10 - 1/(98 - 1/(970 - ...))) and the simple continued fraction expansion 1/(1 + 1/((10-2) + 1/(1 + 1/((98-2) + 1/(1 + 1/((970-2) + 1/(1 + ...))))))).
%F F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((10^2-4) + 1/(1 + 1/((98^2-4) + 1/(1 + 1/((970^2-4) + 1/(1 + ...))))))). Cf. A174503 and A005248. (End)
%F a(-n) = a(n). - _Michael Somos_, Feb 25 2014
%F From _Peter Bala_, Oct 16 2019: (Start)
%F 8*Sum_{n >= 1} 1/(a(n) - 12/a(n)) = 1.
%F 12*Sum_{n >= 1} (-1)^(n+1)/(a(n) + 8/a(n)) = 1.
%F Series acceleration formulas for sums of reciprocals:
%F Sum_{n >= 1} 1/a(n) = 1/8 - 12*Sum_{n >= 1} 1/(a(n)*(a(n)^2 - 12)) and
%F Sum_{n >= 1} (-1)^(n+1)/a(n) = 1/12 + 8*Sum_{n >= 1} (-1)^(n+1)/(a(n)*(a(n)^2 + 8)).
%F Sum_{n >= 1} 1/a(n) = ( (theta_3(5-sqrt(24)))^2 - 1 )/4 and
%F Sum_{n >= 1} (-1)^(n+1)/a(n) = ( 1 - (theta_3(sqrt(24)-5))^2 )/4, where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122). Cf. A153415 and A003499. (End)
%F E.g.f.: 2*exp(5*x)*cosh(2*sqrt(6)*x). - _Stefano Spezia_, Oct 18 2019
%F From _Peter Bala_, Mar 29 2022: (Start)
%F a(n) = 2*T(n,5), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.
%F a(2^n) = A135927(n+1) and a(3^n) = A006242(n+1), both for n >= 0. (End)
%e a(4) = 9602 = 10*a(3) - a(2) = 10*970 - 98 = (5+sqrt(24))^4 + (5-sqrt(24))^4.
%t a[0] = 2; a[1] = 10; a[n_] := 10a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t LinearRecurrence[{10,-1}, {2,10}, 30] (* _G. C. Greubel_, Nov 07 2018 *)
%o (Sage) [lucas_number2(n,10,1) for n in range(27)] # _Zerinvary Lajos_, Jun 25 2008
%o (PARI) polsym(x^2 - 10*x + 1,20) \\ _Charles R Greathouse IV_, Jun 11 2011
%o (PARI) {a(n) = 2 * real( (5 + 2 * quadgen(24))^n )}; /* _Michael Somos_, Feb 25 2014 */
%o (Magma) I:=[2,10]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Nov 07 2018
%Y Cf. A005248, A006242, A036336, A086927, A135927, A174503.
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003
%E More terms from _Colin Barker_, Feb 25 2014
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