OFFSET
0,3
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 (8,-14).
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
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