Search: id:A049684
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%I A049684
%S A049684 0,1,9,64,441,3025,20736,142129,974169,6677056,45765225,313679521,
%T A049684 2149991424,14736260449,101003831721,692290561600,4745030099481,
%U A049684 32522920134769,222915410843904,1527884955772561,10472279279564025
%N A049684 F(2n)^2 where F() = Fibonacci numbers A000045.
%C A049684 This is the r=9 member of the r-family of sequences S_r(n) defined in 
               A092184 where more information can be found.
%D A049684 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 27.
%H A049684 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof 
               of a nonlinearity</a>
%H A049684 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A049684 G.f.: (x+x^2)/((1-x)*(1-7*x+x^2)).
%F A049684 a(n) = 8*a(n-1)-8*a(n-2)+a(n-3), n>2. a(0) = 0, a(1) = 1, a(2) = 9.
%F A049684 a(n) = 7a(n-1)-a(n-2)+2 = A001906(n)^2.
%F A049684 a(n) = 1/5*{-2+[(7+sqrt(45))/2]^n+[(7-sqrt(45))/2]^n}. - R. Stephan, 
               Apr 14 2004
%F A049684 a(n)= 2*(T(n, 7/2)-1)/5 with twice the Chebyshev's polynomials of the 
               first kind evaluated at x=7/2: 2*T(n, 7/2)= A056854(n). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsr\
               uhe_DOT_de), Oct 18 2004
%F A049684 F(2) + F(6) + F(10) +...+ F(4n+2) = F(2n+1)*F(2n+3) - 1.
%o A049684 (PARI) a(n)=fibonacci(2*n)^2
%o A049684 (Mupad) numlib::fibonacci(2*n)^2 $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 13 2008
%o A049684 (Other) sage: [(fibonacci(2*n))^2 for n in xrange(0, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%Y A049684 A049684(n)=(A000032(4*n)-2)/5. First differences give A033890.
%Y A049684 a(n) = A064170(n+2) - 1 = (1/5) A081070. Bisection of A007598 and A064841.
%Y A049684 First differences of A103434.
%Y A049684 Sequence in context: A143631 A083328 A000846 this_sequence A037540 A037484 
               A013566
%Y A049684 Adjacent sequences: A049681 A049682 A049683 this_sequence A049685 A049686 
               A049687
%K A049684 nonn,easy,nice
%O A049684 0,3
%A A049684 Clark Kimberling (ck6(AT)evansville.edu)
%E A049684 Better description and more terms from Michael Somos

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