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A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...). 17
0, 1, 6, 29, 132, 589, 2610, 11537, 50952, 224953, 993054, 4383653, 19350540, 85417669, 377052234, 1664389721, 7346972688, 32431108081, 143157839670, 631929281453, 2789470811028, 12313319895997, 54353623698786 (list; graph; refs; listen; history; text; internal format)
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
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
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
Sequence in context: A351146 A026675 A026873 * A026866 A045445 A026884
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved

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Last modified June 22 14:17 EDT 2024. Contains 373587 sequences. (Running on oeis4.)