%I A029547
%S A029547 1,34,1155,39236,1332869,45278310,1538129671,52251130504,
%T A029547 1775000307465,60297759323306,2048348816684939,69583562007964620,
%U A029547 2363792759454112141,80299370259431848174,2727814796061228725775
%N A029547 Expansion of 1/(1-34*x+x^2).
%C A029547 Chebyshev sequence U(n,17)=S(n,34) with Diophantine property.
%C A029547 b(n)^2 - 2*(12*a(n))^2 = 1 with the companion sequence b(n)=A056771(n+1).
%C A029547 b(n)^2 - 2*(12*a(n))^2 = 1 where the companion sequence b(n)=A056771(n+1). 
               - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 19 2008
%H A029547 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A029547 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A029547 a(n) = 34*a(n-1) - a(n-2), a(-1) = 0, a(0) = 1.
%F A029547 a(n) = S(n, 34) with S(n, x) := U(n, x/2) Chebyshev's polynomials of 
               the 2nd kind. See A049310.
%F A029547 a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap = 17+12*sqrt(2) and am 
               = 17-12*sqrt(2).
%F A029547 a(n) = sum((-1)^k*binomial(n-k, k)*34^(n-2*k), k = 0..floor(n/2)).
%F A029547 a(n) = sqrt((A056771(n+1)^2 -1)/2)/12.
%F A029547 a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3); a(0) = 0, a(1) = 1, a(2) = 34. 
               Also a(n) = (sqrt(2)/48)*((17+12*sqrt(2))^n-(17-12*sqrt(2))^n); a(n) 
               = (sqrt(2)/48)*((3+2*sqrt(2))^2n-(3-2*sqrt(2))^2n); a(n) = (sqrt(2)/
               48)*((1+sqrt(2))^4n-(1-sqrt(2))^4n). - Antonio A. Olivares (olivares14031(AT)yahoo.com), 
               Mar 19 2008
%p A029547 with (combinat):seq(fibonacci(4*n,2)/12, n=1..15); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 21 2008
%t A029547 lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 17]], {n, 0, 8^2}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
%o A029547 (PARI) A029547(n, x=[0,1],A=[17,72*4;1,17]) = vector(n,i,(x*=A)[1]) - 
               M. F. Hasler, May 26 2007
%o A029547 (Other) sage: [lucas_number1(n,34,1) for n in xrange(1, 16)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
%Y A029547 A091761 is an essentially identical sequence.
%Y A029547 Sequence in context: A098607 A075292 A158696 this_sequence A091761 A009978 
               A041545
%Y A029547 Adjacent sequences: A029544 A029545 A029546 this_sequence A029548 A029549 
               A029550
%K A029547 nonn
%O A029547 0,2
%A A029547 N. J. A. Sloane (njas(AT)research.att.com).
%E A029547 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Dec 11 2002