OFFSET
0,2
COMMENTS
a(n) = (t(i+4n+1) - t(i))/(t(i+2n+1) - t(i+2n)), where (t) is any sequence of the form t(n+2) = 4t(n+1) - 4t(n) + t(n-1) or of the form t(n+1) = 3t(n) - t(n-1) without regard to initial values as long as t(i+2n+1) - t(i+2n) != 0. - Klaus Purath, Jun 24 2024
REFERENCES
V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers, A Publication of the Fibonacci Association, Houghton Mifflin Co., 1969, pp. 27-29.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).
FORMULA
a(n) = 7*a(n-1) - a(n-2), with a(0)=1, a(1)=11.
a(n) = (11*(((7+3*sqrt(5))/2)^n - ((7-3*sqrt(5))/2)^n) - (((7+3*sqrt(5))/2)^(n-1) - ((7-3*sqrt(5))/2)^(n-1)))/3*sqrt(5).
G.f.: (1+4*x)/(1-7*x+x^2). - Philippe Deléham, Nov 02 2008
MAPLE
with(combinat); seq(fibonacci(4*n+2)+fibonacci(4*n), n = 0..30); # G. C. Greubel, Jan 16 2020
MATHEMATICA
LucasL[4*Range[0, 30]+1] (* or *) LinearRecurrence[{7, -1}, {1, 11}, 30] (* G. C. Greubel, Dec 24 2017 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1+4*x)/(1-7*x+x^2)) \\ G. C. Greubel, Dec 24 2017
(Magma) [Lucas(4*n+1): n in [0..30]]; // G. C. Greubel, Dec 24 2017
(Sage) [lucas_number2(4*n+1, 1, -1) for n in (0..30)] # G. C. Greubel, Jan 16 2020
(GAP) List([0..30], n-> Lucas(1, -1, 4*n+1)[2] ); # G. C. Greubel, Jan 16 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jul 11 2000
STATUS
approved