|
%I #83 Mar 28 2023 21:52:34
%S 0,3,21,144,987,6765,46368,317811,2178309,14930352,102334155,
%T 701408733,4807526976,32951280099,225851433717,1548008755920,
%U 10610209857723,72723460248141,498454011879264,3416454622906707,23416728348467685,160500643816367088,1100087778366101931,7540113804746346429
%N a(n) = Fibonacci(4n).
%C (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 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 Fibonacci numbers divisible by 3. - _Reinhard Zumkeller_, Aug 20 2011
%H Nathaniel Johnston, <a href="/A033888/b033888.txt">Table of n, a(n) for n = 0..300</a>
%H Piero Filipponi and Marco Bucci, <a href="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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F a(n) = 7*a(n-1) - a(n-2).
%F 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 a(n) = Sum_{k=0..n} F(3n-k)*binomial(n, k). - _Benoit Cloitre_, Jun 07 2004
%F a(n) = Lucas(2n) * Lucas(n) * Fibonacci(n). - _Ralf Stephan_, Sep 25 2004
%F G.f.: 3*x/(1-7*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F a(n) = 3*A004187(n). - _R. J. Mathar_, Sep 03 2010
%F a(n) = Fibonacci[(4*n + 2)] modulo Fibonacci[(4*n + 1)]. - _Artur Jasinski_, Nov 15 2011 (corrected by _Iain Fox_, Dec 18 2017)
%F a(n) = (A337929(n) + A337928(n)) / 2. - _Flávio V. Fernandes_, Feb 06 2021
%F E.g.f.: 2*exp(7*x/2)*sinh(3*sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Feb 07 2021
%F a(n) = Sum_{k>=0} Fibonacci(2*n*k)/Lucas(2*n)^k (Filipponi and Bucci, 1994). - _Amiram Eldar_, Jan 17 2022
%e G.f. = 3*x + 21*x^2 + 144*x^3 + 987*x^4 + 6765*x^5 + 46368*x^6 + ...
%p A033888:=n->combinat[fibonacci](4*n): seq(A033888(n), n=0..30); # _Wesley Ivan Hurt_, Apr 26 2017
%t Table[Fibonacci[4*n], {n,0,14}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)
%t 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 (MuPAD) numlib::fibonacci(n*4) $ n = 0..30; // _Zerinvary Lajos_, May 08 2008
%o (Sage) [lucas_number1(n,3,1)*lucas_number2(n,3,1) for n in range(0,21)] # _Zerinvary Lajos_, Jun 28 2008
%o (Sage) [fibonacci(4*n) for n in range(0, 20)] # _Zerinvary Lajos_, May 15 2009
%o (Magma) [ Fibonacci(4*n): n in [0..100]]; // _Vincenzo Librandi_, Apr 15 2011
%o (PARI) a(n)=fibonacci(4*n) \\ _Charles R Greathouse IV_, Feb 03 2014
%o (PARI) first(n) = Vec(3*x/(1 - 7*x + x^2) + O(x^n), -n) \\ _Iain Fox_, Dec 18 2017
%o (PARI) a(n) = fibonacci(4*n + 2) % fibonacci(4*n + 1) \\ _Iain Fox_, Dec 18 2017
%Y Cf. A000032, A000045, A004187, A033890, A337928, A337929.
%Y Fourth column of array A102310.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
| 