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A049684 a(n) = Fibonacci(2n)^2. 20

%I

%S 0,1,9,64,441,3025,20736,142129,974169,6677056,45765225,313679521,

%T 2149991424,14736260449,101003831721,692290561600,4745030099481,

%U 32522920134769,222915410843904,1527884955772561,10472279279564025

%N a(n) = Fibonacci(2n)^2.

%C 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 Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.

%D Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 27.

%H Vincenzo Librandi, <a href="/A049684/b049684.txt">Table of n, a(n) for n = 0..500</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%H E. Kilic, Y. T. Ulutas, N. Omur, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Omur/omur6.html">A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters</a>, J. Int. Seq. 14 (2011) #11.5.6, table 1, k=2.

%H R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof of a nonlinearity</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8, -8, 1).

%F G.f.: (x+x^2) / ((1-x)*(1-7*x+x^2)).

%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) with n>2, a(0)=0, a(1)=1, a(2)=9.

%F a(n) = 7*a(n-1) - a(n-2) + 2 = A001906(n)^2.

%F a(n) = (A000032(4*n)-2)/5. [This is in Koshy's book (reference under A065563) on p. 88, attributed to Lucas 1876.] - _Wolfdieter Lang_, Aug 27 2012]

%F a(n) = 1/5*(-2 + ( (7+sqrt(45))/2 )^n + ( (7-sqrt(45))/2 )^n). - _Ralf Stephan_, Apr 14 2004

%F 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). - _Wolfdieter Lang_, Oct 18 2004

%F a(n) = F(n-1)*F(n+1)-1. [_Bruno Berselli_, Feb 12 2015]

%F a(n) = Sum_{i=1..n} F(4*i-2) for n>0. [_Bruno Berselli_, Aug 25 2015]

%t Join[{a=0, b=1}, Table[c=7*b-1*a+2; a=b; b=c, {n, 60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)

%t Fibonacci[Range[0, 40, 2]]^2 (* _Harvey P. Dale_, Mar 22 2012 *)

%t Table[Fibonacci[n - 1] Fibonacci[n + 1] - 1, {n, 0, 40, 2}] (* _Bruno Berselli_, Feb 12 2015 *)

%t LinearRecurrence[{8, -8, 1},{0, 1, 9},21] (* _Ray Chandler_, Sep 23 2015 *)

%o (PARI) a(n)=fibonacci(2*n)^2

%o (MuPAD) numlib::fibonacci(2*n)^2 $ n = 0..35; // _Zerinvary Lajos_, May 13 2008

%o (Sage) [fibonacci(2*n)^2 for n in xrange(0, 21)] # _Zerinvary Lajos_, May 15 2009

%Y First differences give A033890.

%Y First differences of A103434.

%Y Bisection of A007598 and A064841.

%Y a(n) = A064170(n+2) - 1 = (1/5) A081070.

%Y Cf. A000045.

%K nonn,nice,easy

%O 0,3

%A _Clark Kimberling_

%E Better description and more terms from _Michael Somos_

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Last modified October 19 07:56 EDT 2018. Contains 316337 sequences. (Running on oeis4.)