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

Author

Paul Barry, Mar 11 2003

Keywords

• nonn
• easy

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).

Crossrefs

Binomial transform of A081182.

Programs

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

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 *)

Sage

[lucas_number1(n,12,34) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

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