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

0,3

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).

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.

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 *)

Prog

(Sage) [lucas_number1(n, 8, 11) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ 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
(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

Crossrefs

Cf. A084326.

Sequence in context: A291662 A110099 A297334 * A252824 A054418 A293115

Adjacent sequences: A091867 A091868 A091869 * A091871 A091872 A091873

Keyword

nonn,easy

Author

Paul Barry, Feb 06 2004

Status

approved