OFFSET
1,4
COMMENTS
This is analogous to the closed form of the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x=(sqrt(2)+1)/2, a(n)/a(n-1) -> x.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
The general form of x is (sqrt(r)+1)/2, r=1,2,3...
a(n) = floor(b(n)/2^n) where b(n) = A052542(n) + 2^(n-1) = 4*b(n-1) - 3*b(n-2) - 2*b(n-3). - R. J. Mathar, Sep 10 2016
MATHEMATICA
Table[Floor[Fibonacci[n, 2]/2^(n-1) +1/2], {n, 1, 50}] (* G. C. Greubel, Oct 02 2018 *)
PROG
(PARI) fib(n, r) = x=(sqrt(r)+1)/2; floor((x^n-(1-x)^n)/sqrt(r)+.5);
g(n, r) = for(m=1, n, print1(fib(m, r)", "));
g(30, 2)
(Magma) [Floor(((1+Sqrt(2))^n - (1-Sqrt(2))^n)/(2^n*Sqrt(2))+ 1/2): n in [2..50]]; // G. C. Greubel, Oct 02 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Apr 01 2008
STATUS
approved