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

0,3

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

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 (6,-7).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A351146 A026675 A026873 * A026866 A045445 A026884

Adjacent sequences: A081176 A081177 A081178 * A081180 A081181 A081182

Keyword

easy,nonn

Author

Paul Barry, Mar 11 2003

Status

approved