%I A054493
%S A054493 1,7,36,175,841,4032,19321,92575,443556,2125207,10182481,48787200,
%T A054493 233753521,1119980407,5366148516,25710762175,123187662361,590227549632,
%U A054493 2827950085801,13549522879375,64919664311076,311048798676007
%N A054493 A Pellian-related recursive sequence.
%C A054493 This is the r=7 member in the r-family of sequences S_r(n+1) defined 
               in A092184 where more information can be found.
%D A054493 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 
               pps. 181-193.
%D A054493 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, 
               pps. 122-125, 194-196.
%D A054493 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 
               7 (1969), pps. 231-242.
%H A054493 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof 
               of a nonlinearity</a>
%H A054493 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A054493 a(n)=5a(n-1)-a(n-2)+2, a(0)=1, a(1)=7.
%F A054493 a(n) = 1/3*{-2+[(5+sqrt(21))/2]^n+[(5-sqrt(21))/2]^n}. - R. Stephan, 
               Apr 14 2004
%F A054493 G.f.: (1+x)/((1-x)*(1-5*x+x^2))=(1+x)/(1-6*x+6*x^2-x^3). From the R. 
               Stephan link.
%F A054493 a(n)= 6*a(n-1)-6*a(n-2)+a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=7.
%F A054493 a(n)=2*T(n, 5/2)-2, with twice the Chebyshev's polynomials of the first 
               kind, 2*T(n, x=5/2)=A003501(n).
%F A054493 a(n)= b(n) + b(n-1), n>=1, with b(n):=A089817(n) the partial sums of 
               S(n, 5)= U(n, 5/2)=A004254(n+1), with S(n, x)=U(n, x/2) Chebyshev's 
               polynomials of the second kind.
%e A054493 A004254 = sqrt{21*(A054493)^2+28*(A054493)}/7.
%Y A054493 Cf. A004254.
%Y A054493 Sequence in context: A102053 A058681 A110310 this_sequence A037538 A037482 
               A147546
%Y A054493 Adjacent sequences: A054490 A054491 A054492 this_sequence A054494 A054495 
               A054496
%K A054493 easy,nonn
%O A054493 0,2
%A A054493 Barry E. Williams, May 06 2000
%E A054493 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 10 2000
%E A054493 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Sep 10 2004