%I #22 Dec 07 2019 12:18:24
%S 2,29,839,24302,703919,20389349,590587202,17106639509,495501958559,
%T 14352450158702,415725552643799,12041688576511469,348793243166188802,
%U 10102962363242963789,292637115290879761079,8476373381072270107502
%N a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.
%C a(n+1)/a(n) converges to ((29+sqrt(837))/2) =28.9654761... Lim a(n)/a(n+1) as n approaches infinity = 0.0345238... =2/(29+sqrt(837)) =(29-sqrt(837))/2. Lim a(n+1)/a(n) as n approaches infinity = 28.9654761... = (29+sqrt(837))/2=2/(29-sqrt(837)). Lim a(n)/a(n+1) = 29 - Lim a(n+1)/a(n).
%C A Chebyshev T-sequence with a Diophantine property.
%C a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 93*(3*b)^2 =+4 with companion sequence b(n)=A097782(n+1), n>=0.
%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%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/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 (29, -1).
%F a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29. a(n) = ((29+sqrt(837))/2)^n + ((29-sqrt(837))/2)^n, (a(n))^2 =a(2n)+2.
%F a(n) = S(n, 29) - S(n-2, 29) = 2*T(n, 29/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
%F a(n) = ap^n + am^n, with ap := (29+3*sqrt(93))/2 and am := (29-3*sqrt(93))/2.
%F G.f.: (2-29*x)/(1-29*x+x^2).
%e a(4) =703919 = 29a(3) - a(2) = 29*24302 - 839= ((29+sqrt(837))/2)^4 + ((29-sqrt(837))/2)^4 = 703918.99999857 + 0.00000142 =703919.
%e (x,y) = (2;0), (29;1), (839;29), (24302,840), ..., give the
%e nonnegative integer solutions to x^2 - 93*(3*y)^2 =+4.
%t a[0] = 2; a[1] = 29; a[n_] := 29a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* _Robert G. Wilson v_, Jan 30 2004 *)
%t LinearRecurrence[{29,-1},{2,29},30] (* _Harvey P. Dale_, May 28 2013 *)
%o (Sage) [lucas_number2(n,29,1) for n in range(0,16)] # _Zerinvary Lajos_, Jun 27 2008
%Y Cf. A083148, A007610.
%Y a(n)=sqrt(4 + 93*(3*A097782(n-1))^2), n>=1.
%Y Cf. A077428, A078355 (Pell +4 equations).
%Y Cf. A090248 for 2*T(n, 27/2).
%K easy,nonn
%O 0,1
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004
%E More terms from _Robert G. Wilson v_, Jan 30 2004
%E Chebyshev and Pell comments from _Wolfdieter Lang_, Aug 31 2004