login
A081180
4th binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
11
0, 1, 8, 50, 288, 1604, 8800, 47944, 260352, 1411600, 7647872, 41420576, 224294400, 1214467136, 6575615488, 35602384000, 192760455168, 1043650265344, 5650555750400, 30593342288384, 165638957801472, 896804870374400
OFFSET
0,3
LINKS
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
FORMULA
a(n) = 8a(n-1) - 14a(n-2), a(0)=0, a(1)=1.
G.f.: x/(1 - 8x + 14x^2).
a(n) = ((4 + sqrt(2))^n - (4 - sqrt(2))^n)/(2*sqrt(2)).
a(n) = Sum_{k=0..n} C(n,2k+1) 2^k*4^(n-2k-1).
If shifted once left, fourth binomial transform of A143095. - Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009, R. J. Mathar, Oct 15 2009
E.g.f.: exp(4*x)*sinh(sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Aug 12 2017
MATHEMATICA
Join[{a=0, b=1}, Table[c=8*b-14*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[x / (1 - 8 x + 14 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8, -14}, {0, 1}, 30] (* Harvey P. Dale, Aug 17 2019 *)
PROG
(Sage) [lucas_number1(n, 8, 14) for n in range(0, 22)] # Zerinvary Lajos, Apr 23 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 8*Self(n-1)-14*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013
CROSSREFS
Binomial transform of A081179.
Cf. A081182.
Sequence in context: A133357 A081675 A283277 * A052177 A127745 A243876
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 11 2003
EXTENSIONS
Modified the completing comment on the fourth binomial transform - R. J. Mathar, Oct 15 2009
STATUS
approved