OFFSET
0,2
COMMENTS
If x=a(n), y=a(n+1), z=a(n+2) or x=a(n+2), y=a(n+1), z=a(n), then x^2 -9*y*x +7*x*z +9*y^2 -9*z*y +z^2 = -45. - Alexander Samokrutov, Jul 02 2015
REFERENCES
Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).
FORMULA
a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
From Paul Barry, Jan 20 2005: (Start)
G.f.: (1-4*x-2*x^2)/(1-8*x+8*x^2-x^3).
a(n) = Sum_{k=0..n} binomial(2n-k, 2k)*2^(2n-3k).
a(n) = 1 - 0^n + Sum_{k=0..2n} binomial(4n-k-1, k). (End)
a(n) = 1-(1/5)*sqrt(5)*{[(7/2)-(3/2)*sqrt(5))^n+[(7/2)+(3/2)*sqrt(5)]^n}, with n>=0. - Paolo P. Lava, Dec 01 2008
MAPLE
with(combinat): for n from 0 to 30 do printf(`%d, `, fibonacci(4*n)+1) od: # James A. Sellers, Mar 01 2003
MATHEMATICA
Fibonacci[4*Range[0, 30]]+1 (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 22}, 30] (* Harvey P. Dale, Feb 26 2015 *)
PROG
(MAGMA) [Fibonacci(4*n)+1: n in [0..30]]; // Vincenzo Librandi, Apr 20 2011
(PARI) Vec((1-4*x-2*x^2)/((1-x)*(1-7*x+x^2)) + O(x^30)) \\ Colin Barker, Dec 23 2014
(Sage) [fibonacci(4*n)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019
(GAP) List([0..30], n-> Fibonacci(4*n)+1); # G. C. Greubel, Jul 15 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. K. Guy, Mar 01 2003
EXTENSIONS
More terms from James A. Sellers, Mar 01 2003
STATUS
proposed