Search: id:A090251
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%I A090251
%S A090251 2,29,839,24302,703919,20389349,590587202,17106639509,495501958559,
%T A090251 14352450158702,415725552643799,12041688576511469,348793243166188802,
%U A090251 10102962363242963789,292637115290879761079,8476373381072270107502
%N A090251 a(n) =29a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 29.
%C A090251 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 A090251 A Chebyshev T-sequence with a Diophantine property.
%C A090251 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 A090251 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 
               (Sec. 30, Satz 3.35, p. 109 and table p. 108).
%H A090251 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A090251 <a href="Sindx_Rea.html#recur1">Index entries for recurrences a(n) = 
               k*a(n - 1) +/- a(n - 2)</a>
%H A090251 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A090251 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 A090251 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 A090251 a(n) = ap^n + am^n, with ap := (29+3*sqrt(93))/2 and am := (29-3*sqrt(93))/
               2.
%F A090251 G.f.: (2-29*x)/(1-29*x+x^2).
%e A090251 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 A090251 (x,y) = (2;0), (29;1), (839;29), (24302,840), ..., give the
%e A090251 nonnegative integer solutions to x^2 - 93*(3*y)^2 =+4.
%t A090251 a[0] = 2; a[1] = 29; a[n_] := 29a[n - 1] - a[n - 2]; Table[ a[n], {n, 
               0, 15}] (from Robert G. Wilson v Jan 30 2004)
%o A090251 sage: [lucas_number2(n,29,1) for n in xrange(0,16)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 27 2008
%Y A090251 Cf. A083148, A007610.
%Y A090251 a(n)=sqrt(4 + 93*(3*A097782(n-1))^2), n>=1.
%Y A090251 Cf. A077428, A078355 (Pell +4 equations).
%Y A090251 Cf. A090248 for 2*T(n, 27/2).
%Y A090251 Sequence in context: A104535 A013517 A006988 this_sequence A087281 A024234 
               A077282
%Y A090251 Adjacent sequences: A090248 A090249 A090250 this_sequence A090252 A090253 
               A090254
%K A090251 easy,nonn
%O A090251 0,1
%A A090251 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004
%E A090251 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 30 2004
%E A090251 Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Aug 31 2004

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