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A033888 a(n) = Fibonacci(4*n).

%I A033888 #94 May 28 2025 00:54:01
%S A033888 0,3,21,144,987,6765,46368,317811,2178309,14930352,102334155,
%T A033888 701408733,4807526976,32951280099,225851433717,1548008755920,
%U A033888 10610209857723,72723460248141,498454011879264,3416454622906707,23416728348467685,160500643816367088,1100087778366101931,7540113804746346429
%N A033888 a(n) = Fibonacci(4*n).
%C A033888 (x,y)=(a(n),a(n+1)) are solutions of (x+y)^2/(1+xy)=9, the other solutions are in A033890. - _Floor van Lamoen_, Dec 10 2001
%C A033888 Sequence A033888 provides half of the solutions to the equation 5*x^2 + 4 is a square. The other half are found in A033890. Lim_{n->infinity} a(n)/a(n-1) = phi^4 = (7+3*sqrt(5))/2. - _Gregory V. Richardson_, Oct 13 2002
%C A033888 Fibonacci numbers divisible by 3. - _Reinhard Zumkeller_, Aug 20 2011
%H A033888 Nathaniel Johnston, <a href="/A033888/b033888.txt">Table of n, a(n) for n = 0..300</a>
%H A033888 Piero Filipponi and Marco Bucci, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-3/filipponi.pdf">On the Integrity of Certain Fibonacci Sums</a>, The Fibonacci Quarterly, Vol. 32, No. 3 (1994), pp. 245-252.
%H A033888 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A033888 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F A033888 a(n) = 7*a(n-1) - a(n-2).
%F A033888 a(n) = ((7+3*sqrt(5))^(n-1) - (7-3*sqrt(5))^(n-1)) / ((2^(n-1))*sqrt(5)). - _Gregory V. Richardson_, Oct 13 2002
%F A033888 a(n) = Sum_{k=0..n} F(3*n-k)*binomial(n, k). - _Benoit Cloitre_, Jun 07 2004
%F A033888 a(n) = Lucas(2*n) * Lucas(n) * Fibonacci(n). - _Ralf Stephan_, Sep 25 2004
%F A033888 G.f.: 3*x/(1-7*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F A033888 a(n) = 3*A004187(n). - _R. J. Mathar_, Sep 03 2010
%F A033888 a(n) = Fibonacci[(4*n + 2)] modulo Fibonacci[(4*n + 1)]. - _Artur Jasinski_, Nov 15 2011 (corrected by _Iain Fox_, Dec 18 2017)
%F A033888 a(n) = (A337929(n) + A337928(n)) / 2. - _Flávio V. Fernandes_, Feb 06 2021
%F A033888 E.g.f.: 2*exp(7*x/2)*sinh(3*sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Feb 07 2021
%F A033888 a(n) = Sum_{k>=0} Fibonacci(2*n*k)/Lucas(2*n)^k (Filipponi and Bucci, 1994). - _Amiram Eldar_, Jan 17 2022
%F A033888 From _Peter Bala_, May 22 2025: (Start)
%F A033888 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/3 (telescoping series: 3/(a(n) - 1/a(n)) = 1/A033890(n) + 1/A033890(n-1)).
%F A033888 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(5) (telescoping product: ((a(n) + 1)/(a(n) - 1))^2 = (5 - 4/Fibonacci(2*n+1)^2)/(5 - 4/Fibonacci(2*n-1)^2) from which we get Product_{k = 1..n} ((a(k) + 1)/(a(k) - 1))^2 = 5 - 4/Fibonacci(2*n+1)^2). (End)
%e A033888 G.f. = 3*x + 21*x^2 + 144*x^3 + 987*x^4 + 6765*x^5 + 46368*x^6 + ...
%p A033888 A033888:=n->combinat[fibonacci](4*n): seq(A033888(n), n=0..30); # _Wesley Ivan Hurt_, Apr 26 2017
%t A033888 Table[Fibonacci[4*n], {n,0,14}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)
%t A033888 Table[Mod[Fibonacci[(4 n + 2)] , Fibonacci[(4 n + 1)]], {n, 1, 10}] (* _Artur Jasinski_, Nov 15 2011 (corrected by _Iain Fox_, Dec 18 2017) *)
%o A033888 (MuPAD) numlib::fibonacci(n*4) $ n = 0..30; // _Zerinvary Lajos_, May 08 2008
%o A033888 (Sage) [lucas_number1(n,3,1)*lucas_number2(n,3,1) for n in range(0,21)] # _Zerinvary Lajos_, Jun 28 2008
%o A033888 (Sage) [fibonacci(4*n) for n in range(0, 20)] # _Zerinvary Lajos_, May 15 2009
%o A033888 (Magma) [ Fibonacci(4*n): n in [0..100]]; // _Vincenzo Librandi_, Apr 15 2011
%o A033888 (PARI) a(n)=fibonacci(4*n) \\ _Charles R Greathouse IV_, Feb 03 2014
%o A033888 (PARI) first(n) = Vec(3*x/(1 - 7*x + x^2) + O(x^n), -n) \\ _Iain Fox_, Dec 18 2017
%o A033888 (PARI) a(n) = fibonacci(4*n + 2) % fibonacci(4*n + 1) \\ _Iain Fox_, Dec 18 2017
%Y A033888 Cf. A000032, A000045, A004187, A033890, A337928, A337929.
%Y A033888 Fourth column of array A102310.
%K A033888 nonn,easy
%O A033888 0,2
%A A033888 _N. J. A. Sloane_