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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved