A093145 Third binomial transform of Fibonacci(3n)/Fibonacci(3).

0, 1, 10, 80, 600, 4400, 32000, 232000, 1680000, 12160000, 88000000, 636800000, 4608000000, 33344000000, 241280000000, 1745920000000, 12633600000000, 91417600000000, 661504000000000, 4786688000000000, 34636800000000000

Offset: 0

Author

Paul Barry, Mar 26 2004

Keywords

• easy
• nonn

Comments

Fifth binomial transform of 1,5,5,25,25,125. - Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
• Index entries for linear recurrences with constant coefficients, signature (10,-20).

Crossrefs

Cf. A000045, A091870.

Programs

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

Mathematica

Join[{a=0,b=1},Table[c=10*b-20*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(5 + s)^n + (1 - s)(5 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
a[n_] := 10 a[n - 1] - 20 a[n - 2]; a[0] = 0; a[1] = 1; Array[a, 22, 0] (* Or *)
CoefficientList[Series[x/(1 - 10 x + 20 x^2), {x, 0, 21}], x] (* Robert G. Wilson v, Mar 07 2011 *)
LinearRecurrence[{10,-20},{0,1},30] (* Harvey P. Dale, Jan 23 2019 *)

Sage

[lucas_number1(n,10,20) for n in range(0, 21)] # Zerinvary Lajos, Apr 26 2009

Formula

G.f.: x/(1 - 10*x + 20*x^2).

a(n) = ((5+sqrt(5))^n - (5-sqrt(5))^n)/(2*sqrt(5)).

a(n) = Sum_{k=0..n} binomial(n, 2*k+ 1)*5^(n-k-1).

a(n) = 10*a(n-1) - 20*a(n-2), n > 1; a(0)=0, a(1)=1. - Zerinvary Lajos, Apr 26 2009

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