|
%I #50 Dec 18 2023 12:16:49
%S 0,1,5,29,165,941,5365,30589,174405,994381,5669525,32325149,184303845,
%T 1050819821,5991314485,34159851709,194764516485,1110461989261,
%U 6331368012245,36098688018269,205818912140325,1173489312774701,6690722212434805
%N Expansion of x/(1 - 5*x - 4*x^2).
%C First differences give A122690(n) = {1, 4, 24, 136, 776, 4424, 25224, ...}. Partial sums of a(n) are {0, 1, 6, 35, 200, ...} = (A123270(n) - 1)/8. - _Alexander Adamchuk_, Nov 03 2006
%C For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 5's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 19 2011
%C Pisano period lengths: 1, 1, 8, 1, 4, 8, 48, 1, 24, 4, 40, 8, 42, 48, 8, 2, 72, 24, 360, 4, ... - _R. J. Mathar_, Aug 10 2012
%H Vincenzo Librandi, <a href="/A015537/b015537.txt">Table of n, a(n) for n = 0..1000</a>
%H Lucyna Trojnar-Spelina and Iwona Włoch, <a href="https://doi.org/10.1007/s40995-019-00757-7">On Generalized Pell and Pell-Lucas Numbers</a>, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,4).
%F a(n) = 5*a(n-1) + 4*a(n-2).
%F a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*4^k*5^(n-2*k-1). - _Paul Barry_, Apr 23 2005
%F a(n) = Sum_{k=0..(n-1)} A122690(k). - _Alexander Adamchuk_, Nov 03 2006
%F a(n) = 2^(n-1)*Fibonacci(n, 5/2) = (2/i)^(n-1)*ChebyshevU(n-1, 5*i/4). - _G. C. Greubel_, Dec 26 2019
%p seq( simplify((2/I)^(n-1)*ChebyshevU(n-1, 5*I/4)), n=0..20); # _G. C. Greubel_, Dec 26 2019
%t LinearRecurrence[{5,4}, {0,1}, 30] (* _Vincenzo Librandi_, Nov 12 2012 *)
%t Table[2^(n-1)*Fibonacci[n, 5/2], {n, 0, 30}] (* _G. C. Greubel_, Dec 26 2019 *)
%o (Sage) [lucas_number1(n,5,-4) for n in range(0, 22)] # _Zerinvary Lajos_, Apr 24 2009
%o (Magma) [n le 2 select n-1 else 5*Self(n-1)+4*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 12 2012
%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-5*x-4*x^2))) \\ _G. C. Greubel_, Jan 01 2018
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # _G. C. Greubel_, Dec 26 2019
%Y Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A122690, A123270, A180222, A180226.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
|
