A091870 A trinomial transform of Fibonacci(3n).

0, 1, 8, 53, 336, 2105, 13144, 81997, 511392, 3189169, 19888040, 124023461, 773419248, 4823095913, 30077155576, 187563189565, 1169656805184, 7294059356257, 45486249993032, 283655347025429, 1768894026280080

Offset: 0

Author

Paul Barry, Feb 06 2004

Keywords

• nonn
• easy

Comments

Binomial transform of A084326.

Second binomial transform of A001076(n) = Fibonacci(3n)/2.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1258
• 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 (8,-11).

Crossrefs

Cf. A084326.

Programs

GAP

a:=[0,1];; for n in [3..30] do a[n]:=8*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Magma

[n le 2 select n-1 else 8*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019

Mathematica

LinearRecurrence[{8, -11}, {0, 1}, 30] (* G. C. Greubel, May 21 2019 *)
CoefficientList[Series[x/(1 - 8 x + 11 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 22 2017 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ G. C. Greubel, May 21 2019

Sage

[lucas_number1(n,8,11) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1 - 8*x + 11*x^2).

a(n) = sqrt(5) * ((4+sqrt(5))^n - (4-sqrt(5))^n) / 10.

a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!*j!*(n-i-j)!)) * Fibonacci(3*i) / 2.