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%I A087800 #41 Sep 08 2022 08:45:11
%S A087800 2,12,142,1692,20162,240252,2862862,34114092,406506242,4843960812,
%T A087800 57721023502,687808321212,8195978831042,97663937651292,
%U A087800 1163771272984462,13867591338162252,165247324784962562,1969100306081388492
%N A087800 a(n) = 12*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12.
%C A087800 a(n+1)/a(n) converges to (6+sqrt(35)) = 11.9160797... a(0)/a(1)=2/12; a(1)/a(2)=12/142; a(2)/a(3)=142/1692; a(3)/a(4)=1692/20162; ... etc. Lim_{n->infinity} a(n)/a(n+1) = 0.0839202... = 1/(6+sqrt(35)) = (6-sqrt(35)).
%C A087800 Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 140 = 0. - _Colin Barker_, Feb 25 2014
%H A087800 Vincenzo Librandi, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwODc4MDAvYjA4NzgwMC50eHQ">Table of n, a(n) for n = 0..200</a>
%H A087800 Tanya Khovanova, <a href="/load/view.php?a=aHR0cDovL3d3dy50YW55YWtob3Zhbm92YS5jb20vUmVjdXJzaXZlU2VxdWVuY2VzL1JlY3Vyc2l2ZVNlcXVlbmNlcy5odG1s">Recursive Sequences</a>
%H A087800 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYSNyZWN1cjE">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A087800 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wMg">Index entries for linear recurrences with constant coefficients</a>, signature (12,-1).
%F A087800 a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
%F A087800 a(n) = 2*A023038(n).
%F A087800 G.f.: (2-12*x)/(1-12*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F A087800 a(-n) = a(n). - _Michael Somos_, May 28 2014
%e A087800 a(4) = 20162 = 12a(3) - a(2) = 12*1692 - 142 = (6+sqrt(35))^4 + (6-sqrt(35))^4 = 20161.9999504 + 0.00004959 = 20162.
%e A087800 G.f. = 2 + 12*x + 142*x^2 + 1692*x^3 + 20162*x^4 + 240252*x^5 + ...
%t A087800 a[0] = 2; a[1] = 12; a[n_] := 12a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t A087800 CoefficientList[Series[(2 - 12 x)/(1 - 12 x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 28 2014 *)
%t A087800 a[ n_] := 2 ChebyshevT[ n, 6]; (* _Michael Somos_, May 28 2014 *)
%t A087800 LinearRecurrence[{12,-1},{2,12},20] (* _Harvey P. Dale_, Jan 29 2019 *)
%o A087800 (Sage) [lucas_number2(n,12,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o A087800 (PARI) Vec((2-12*x)/(1-12*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 25 2014
%o A087800 (PARI) {a(n) = 2 * polchebyshev( n, 1, 6)}; /* _Michael Somos_, May 28 2014 */
%o A087800 (Magma) I:=[2,12]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Nov 07 2018
%Y A087800 Cf. A009747, A086928, A001927, A023038.
%K A087800 easy,nonn
%O A087800 0,1
%A A087800 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003

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