A102312 - OEIS

%I #43 Jan 05 2025 19:51:38

%S 0,5,55,610,6765,75025,832040,9227465,102334155,1134903170,

%T 12586269025,139583862445,1548008755920,17167680177565,

%U 190392490709135,2111485077978050,23416728348467685,259695496911122585,2880067194370816120,31940434634990099905

%N a(n) = Fibonacci(5*n).

%H Colin Barker, <a href="/A102312/b102312.txt">Table of n, a(n) for n = 0..950</a>

%H Michael D.Hirschhorn, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/51-3/HirschhornNaiveProof.pdf">A Naive Proof that F5n = 0 (mod 5)</a>, Fib. Q. 51(3), 2013, 256-258.

%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 (11,1).

%F G.f.: 5*x/(1-11*x-x^2).

%F a(n) = A000045(5*n) = 5*A049666(n).

%F a(n) = Fibonacci(2*n)*Lucas(3*n)+Fibonacci(n). Lucas =A000032(n), Fibonacci=A000045(n). - _Gary Detlefs_, Dec 22 2012

%F a(n) = (-((11 - 5*sqrt(5))/2)^n + ((11+5*sqrt(5))/2)^n)/sqrt(5). - _Colin Barker_, Nov 10 2016

%F a(n) = 11*a(n-1)+a(n-2). - _Mike Speciner_, Aug 20 2024

%p seq(combinat:-fibonacci(5*n), n=0..100); # _Robert Israel_, Dec 12 2014

%t Table[ Fibonacci[5n], {n, 0, 17}] (* _Robert G. Wilson v_, Jan 09 2005 *)

%o (Sage) [fibonacci(5*n) for n in range(0, 18)] # _Zerinvary Lajos_, May 15 2009

%o (Magma) [Fibonacci(5*n): n in [0..100]]; // _Vincenzo Librandi_, Apr 17 2011

%o (PARI) vector(18,n,fibonacci(5*n)) \\ _Edward Jiang_, Dec 11 2014

%o (PARI) concat(0, Vec(5*x/(1-11*x-x^2) + O(x^30))) \\ _Colin Barker_, Nov 10 2016

%Y Essentially the fifth column of array A102310.

%Y Cf. A049666. [_Zerinvary Lajos_, May 15 2009]

%Y Cf. A138134 (partial sums).

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, Jan 06 2005