|
%I #65 Dec 27 2023 11:53:51
%S 2,13,89,610,4181,28657,196418,1346269,9227465,63245986,433494437,
%T 2971215073,20365011074,139583862445,956722026041,6557470319842,
%U 44945570212853,308061521170129,2111485077978050
%N a(n) = Fibonacci(4*n+3).
%H G. C. Greubel, <a href="/A033891/b033891.txt">Table of n, a(n) for n = 0..1000</a>
%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). - _Floor van Lamoen_, Dec 10 2001
%F G.f.: (2-x)/(1-7*x+x^2). - _Philippe Deléham_, Nov 17 2008
%F a(n) = A167816(4*n+3). - _Reinhard Zumkeller_, Nov 13 2009
%F a(n) = Fibonacci(2*n+2)^2 + Fibonacci(2*n+1)^2. - _Gary Detlefs_, Oct 12 2011
%F a(n) = 2*A004187(n+1) - A004187(n). - _R. J. Mathar_, Nov 26 2011
%F a(n) = A004187(n+1) + A049685(n). - _Yuriy Sibirmovsky_, Sep 15 2016
%F From _Peter Bala_, Aug 11 2022: (Start)
%F Let n ** m = n*m + floor(phi*n)*floor(phi*m), where phi = (1 + sqrt(5))/2, denote the Porta-Stolarsky star product of the integers n and m (see A101858). Then a(n) = 2 ** 2 ** ... ** 2 (n+1 factors).
%F a(2*n+1) = a(n) ** a(n) = Fibonacci(8*n+7); a(3*n+2) = a(n) ** a(n) ** a(n) = Fibonacci(12*n+11) and so on. (End)
%t Table[Fibonacci[4*n+3],{n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)
%t LinearRecurrence[{7,-1},{2,13},31] (* or *) CoefficientList[Series[ (2-x)/(1-7x+x^2),{x,0,30}],x] (* _Harvey P. Dale_, May 03 2011 *)
%o (Magma) [Fibonacci(4*n+3): n in [0..30]]; // _Vincenzo Librandi_, Apr 17 2011
%o (PARI) a(n)=fibonacci(4*n+3) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Sage) [fibonacci(4*n+3) for n in (0..30)] # _G. C. Greubel_, Jul 14 2019
%o (GAP) List([0..30], n-> Fibonacci(4*n+3)); # _G. C. Greubel_, Jul 14 2019
%Y Cf. A000045, A004187, A049685, A101858, A167816.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
| 