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a(n) = 8*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8.
4

%I #43 Mar 23 2024 12:20:00

%S 2,8,62,488,3842,30248,238142,1874888,14760962,116212808,914941502,

%T 7203319208,56711612162,446489578088,3515205012542,27675150522248,

%U 217885999165442,1715412842801288,13505416743244862,106327921103157608

%N a(n) = 8*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 8.

%C a(n+1)/a(n) converges to (4+sqrt(15)) = 7.872983... a(0)/a(1)=2/8; a(1)/a(2)=8/62; a(2)/a(3)=62/488; a(3)/a(4)=488/3842; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.127016... = 1/(4+sqrt(15)) = (4-sqrt(15)).

%C Twice A001091. - _John W. Layman_, Sep 25 2003

%C Except for the first term, positive values of x (or y) satisfying x^2 - 8xy + y^2 + 60 = 0. - _Colin Barker_, Feb 13 2014

%H Vincenzo Librandi, <a href="/A086903/b086903.txt">Table of n, a(n) for n = 0..200</a>

%H P. Bala, <a href="/A174500/a174500_2.pdf">Some simple continued fraction expansions for an infinite product, Part 1</a>

%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 (8,-1).

%F a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.

%F G.f.: (2-8*x)/(1-8*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 = 4 - sqrt(15). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.12474 84992 41370 33639 ... = 2 + 1/(8 + 1/(62 + 1/(488 + ...))). Cf. A174502 and A005248.

%F Also F(-alpha) = 0.87474 74663 84045 35032 ... has the continued fraction representation 1 - 1/(8 - 1/(62 - 1/(488 - ...))) and the simple continued fraction expansion 1/(1 + 1/((8-2) + 1/(1 + 1/((62-2) + 1/(1 + 1/((488-2) + 1/(1 + ...))))))).

%F F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((8^2-4) + 1/(1 + 1/((62^2-4) + 1/(1 + 1/((488^2-4) + 1/(1 + ...))))))).

%F (End)

%e a(4) = 3842 = 8*a(3) - a(2) = 8*488 - 62 = (4+sqrt(15))^4 + (4-sqrt(15))^4 = 3841.9997397 + 0.0002603 = 3842.

%t a[0] = 2; a[1] = 8; a[n_] := 8a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 19}] (* _Robert G. Wilson v_, Jan 30 2004 *)

%t CoefficientList[Series[(2 - 8 x)/(1 - 8 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 15 2014 *)

%t LinearRecurrence[{8,-1},{2,8},30] (* _Harvey P. Dale_, Jan 18 2015 *)

%o (Sage) [lucas_number2(n,8,1) for n in range(27)] # _Zerinvary Lajos_, Jun 25 2008

%o (Magma) I:=[2,8]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2014

%Y Cf. A086594, A058316, A006245, A009271, A005248, A174502.

%K easy,nonn

%O 0,1

%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21 2003