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%I #61 Sep 25 2024 01:53:04
%S 0,8,144,2584,46368,832040,14930352,267914296,4807526976,86267571272,
%T 1548008755920,27777890035288,498454011879264,8944394323791464,
%U 160500643816367088,2880067194370816120,51680708854858323072,927372692193078999176,16641027750620563662096
%N a(n) = Fibonacci(6*n).
%C All terms are divisible by 8. - _Alonso del Arte_, Jul 27 2013
%C Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci numbers which are the sum of the three numbers of a Pythagorean triple (checked up to F(80)). - _Felix Huber_, Nov 03 2023
%H Colin Barker, <a href="/A134492/b134492.txt">Table of n, a(n) for n = 0..500</a>
%H Hacène Belbachir, Soumeya Merwa Tebtoub and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%F a(n) = 18*a(n-1) - a(n-2) = 8*A049660(n). G.f.: 8*x/(1-18*x+x^2). - _R. J. Mathar_, Feb 16 2010
%F a(n) = A000045(A008588(n)). - _Michel Marcus_, Nov 08 2013
%F a(n) = ((-1+(9+4*sqrt(5))^(2*n)))/(sqrt(5)*(9+4*sqrt(5))^n). - _Colin Barker_, Jan 24 2016
%F a(n) = L(2n-1) * F(2n+1)^2 + L(2n+1) * F(2n-1)^2, where F(n) = A000045(n) and L(n) = A000032(n). - _Diego Rattaggi_, Nov 12 2020
%F a(n) = Fibonacci(3*n) * Lucas(3*n) = A000045(3*n) * A000032(3*n) = A014445(n) * A014448(n). - _Amiram Eldar_, Jan 11 2022
%t Table[Fibonacci[6n], {n, 0, 30}]
%t LinearRecurrence[{18,-1},{0,8},30] (* _Harvey P. Dale_, Aug 15 2017 *)
%o (MuPAD) numlib::fibonacci(6*n) $ n = 0..25; // _Zerinvary Lajos_, May 09 2008
%o (Sage) [fibonacci(6*n) for n in range(0, 17)] # _Zerinvary Lajos_, May 15 2009
%o (Magma) [Fibonacci(6*n): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011
%o (PARI) a(n)=fibonacci(6*n) \\ _Charles R Greathouse IV_, Sep 16 2015
%o (PARI) concat(0, Vec(8*x/(1-18*x+x^2) + O(x^20))) \\ _Colin Barker_, Jan 24 2016
%Y Cf. A000032, A000045, A008588, A049660, A079343, A014445, A014448, A134493, A134494, A134495, A103134, A134497, A134498.
%K nonn,easy
%O 0,2
%A _Artur Jasinski_, Oct 28 2007
%E Offset corrected by _R. J. Mathar_, Feb 16 2010