|
%I #40 Sep 08 2022 08:45:13
%S 0,1,10,80,600,4400,32000,232000,1680000,12160000,88000000,636800000,
%T 4608000000,33344000000,241280000000,1745920000000,12633600000000,
%U 91417600000000,661504000000000,4786688000000000,34636800000000000
%N Third binomial transform of Fibonacci(3n)/Fibonacci(3).
%C Fifth binomial transform of 1,5,5,25,25,125. - Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009
%H Vincenzo Librandi, <a href="/A093145/b093145.txt">Table of n, a(n) for n = 0..200</a>
%H S. Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-20).
%F G.f.: x/(1 - 10*x + 20*x^2).
%F a(n) = ((5+sqrt(5))^n - (5-sqrt(5))^n)/(2*sqrt(5)).
%F a(n) = Sum_{k=0..n} binomial(n, 2*k+ 1)*5^(n-k-1).
%F a(n) = 10*a(n-1) - 20*a(n-2), n > 1; a(0)=0, a(1)=1. - _Zerinvary Lajos_, Apr 26 2009
%F G.f.: A(x) = x*G(0)/(1-5*x) where G(k) = 1 + 5*x/(1-5*x - x*(1-5*x)/(x + (1-5*x)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 30 2012
%t Join[{a=0,b=1},Table[c=10*b-20*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 27 2011 *)
%t f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(5 + s)^n + (1 - s)(5 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
%t a[n_] := 10 a[n - 1] - 20 a[n - 2]; a[0] = 0; a[1] = 1; Array[a, 22, 0] (* Or *)
%t CoefficientList[Series[x/(1 - 10 x + 20 x^2), {x, 0, 21}], x] (* _Robert G. Wilson v_, Mar 07 2011 *)
%t LinearRecurrence[{10,-20},{0,1},30] (* _Harvey P. Dale_, Jan 23 2019 *)
%o (Sage) [lucas_number1(n,10,20) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 26 2009
%o (Magma) [n le 2 select n - 1 else 10*Self(n-1)-20*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Dec 30 2012
%Y Cf. A000045, A091870.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Mar 26 2004
|
