A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).

0, 1, 6, 29, 132, 589, 2610, 11537, 50952, 224953, 993054, 4383653, 19350540, 85417669, 377052234, 1664389721, 7346972688, 32431108081, 143157839670, 631929281453, 2789470811028, 12313319895997, 54353623698786

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• easy
• nonn

Comments

Binomial transform of 0, 1, 4, 14, 48, ... (A007070 with offset 1) and second binomial transform of A000129. - R. J. Mathar, Dec 10 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Sergio Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
• Index entries for linear recurrences with constant coefficients, signature (6,-7).

Crossrefs

Cf. A081180, A081182, A081183, A081184, A081185, A153593.

Programs

Magma

I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013

Maple

f:= gfun:-rectoproc({a(n) = 6*a(n-1)-7*a(n-2), a(0)=0, a(1)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 15 2016

Mathematica

CoefficientList[Series[x/(1-6 x +7 x^2), {x,0,30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{6,-7}, {0,1}, 41] (* G. C. Greubel, Jan 14 2024 *)

Sage

[lucas_number1(n,6,7) for n in range(0, 23)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = 6*a(n-1) - 7*a(n-2), a(0)=0, a(1)=1.

G.f.: x/(1-6*x+7*x^2).

a(n) = ((3+sqrt(2))^n - (3-sqrt(2))^n)/(2*sqrt(2)). [Corrected by Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008]

a(n) = 3^(n-1) Sum_{i>=0} binomial(n, 2i+1) * (2/9)^i. - Sergio Falcon, Mar 15 2016

a(n) = 2^(-1/2)*7^(n/2)*sinh(n*arcsinh(sqrt(2/7))). - Robert Israel, Mar 15 2016

E.g.f.: exp(3*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017

a(n) = 7^((n-1)/2)*ChebyshevU(n-1, 3/sqrt(7)). - G. C. Greubel, Jan 14 2024