A081183 6th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 12, 110, 912, 7204, 55440, 420344, 3159168, 23618320, 176008128, 1309074656, 9724619520, 72186895936, 535605687552, 3972913788800, 29464372088832, 218493396246784, 1620132103941120, 12012809774902784, 89069225764835328

Offset

0,3

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 (12,-34).

Formula

a(n) = 12*a(n-1) - 34*a(n-2) with n > 1, a(0)=0, a(1)=1.

G.f.: x/(1 - 12*x + 34*x^2).

a(n) = ((6 + sqrt(2))^n - (6 - sqrt(2))^n)/(2*sqrt(2)).

a(n) = Sum_{k=0..n} C(n,2k+1)*2^k*6^(n-2k-1).

E.g.f.: exp(6*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

Mathematica

CoefficientList[Series[x / (1 - 12 x + 34 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12, -34}, {0, 1}, 30] (* Harvey P. Dale, Jul 31 2025 *)

Prog

(SageMath) [lucas_number1(n, 12, 34) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) [n le 2 select n-1 else 12*Self(n-1)-34*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Aug 07 2013

Crossrefs

Binomial transform of A081182.

Sequence in context: A037581 A177071 A383457 * A069294 A000559 A037714

Adjacent sequences: A081180 A081181 A081182 * A081184 A081185 A081186

Keyword

nonn,easy

Author

Paul Barry, Mar 11 2003

Status

approved