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A014729 Squares of even Fibonacci numbers.

%I A014729 #48 Jul 02 2025 16:01:55
%S A014729 0,4,64,1156,20736,372100,6677056,119814916,2149991424,38580030724,
%T A014729 692290561600,12422650078084,222915410843904,4000054745112196,
%U A014729 71778070001175616,1288005205276048900,23112315624967704576,414733676044142633476,7442093853169599697984
%N A014729 Squares of even Fibonacci numbers.
%H A014729 Colin Barker, <a href="/A014729/b014729.txt">Table of n, a(n) for n = 0..700</a>
%H A014729 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (17,17,-1)
%F A014729 a(n) = (1/5)*(Fibonacci(6*n+3) - 2*Fibonacci(6*n) - 2*(-1)^n). - _Ralf Stephan_, May 14 2004
%F A014729 G.f.: 4*(-x^2+x)/((1+x)*(1-18*x+x^2)). - _Ralf Stephan_, May 14 2004
%F A014729 a(n) = Fibonacci(3*n)^2. - _Gary Detlefs_, Nov 28 2010
%F A014729 a(n) = (-1)^(n+1)*(Fibonacci(n)*Fibonacci(7*n)-Fibonacci(4*n)^2). - _Gary Detlefs_, Nov 28 2010
%F A014729 a(n) = (-2*(-1)^n+(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n)/5. - _Colin Barker_, Mar 04 2016
%F A014729 a(n) = A014445(n)^2. - _Sean A. Irvine_, Nov 18 2018
%t A014729 (Table[Fibonacci@ n, {n, 0, 55}] /. n_ /; OddQ@ n -> Nothing)^2 (* or *)
%t A014729 CoefficientList[Series[4 (-x^2 + x)/((1 + x) (1 - 18 x + x^2)), {x, 0, 18}], x] (* _Michael De Vlieger_, Mar 04 2016 *)
%t A014729 LinearRecurrence[{17,17,-1},{0,4,64},20] (* _Harvey P. Dale_, Aug 02 2024 *)
%o A014729 (MuPAD) numlib::fibonacci(3*n)^2 $ n = 0..25; // _Zerinvary Lajos_, May 09 2008
%o A014729 (Sage) [fibonacci(3*n)^2 for n in range(0, 17)] # _Zerinvary Lajos_, May 15 2009
%o A014729 (PARI) concat(0, Vec(4*x*(1-x)/((1+x)*(1-18*x+x^2)) + O(x^40))) \\ _Colin Barker_, Mar 04 2016
%o A014729 (Magma) [Fibonacci(3*n)^2: n in [0..20]]; // _Vincenzo Librandi_, Nov 19 2018
%Y A014729 Cf. A014445.
%K A014729 nonn,easy
%O A014729 0,2
%A A014729 _Mohammad K. Azarian_
%E A014729 More terms from _James Sellers_