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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 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

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