A136422 Floor((x^n - (1-x)^n)/sqrt(3)+.5) where x = (sqrt(3)+1)/2.

1, 1, 2, 2, 3, 4, 5, 7, 10, 13, 18, 24, 33, 45, 62, 85, 116, 158, 216, 296, 404, 551, 753, 1029, 1406, 1920, 2623, 3583, 4895, 6687, 9134, 12477, 17044, 23283, 31805, 43447, 59349, 81072, 110747, 151283, 206657, 282298, 385626, 526775, 719589, 982976

Offset

1,3

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(3)+1)/2, a(n)/a(n-1) -> x.

Links

Table of n, a(n) for n=1..46.

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) = 2*A002605(n)+2^(n-1). - R. J. Mathar, Sep 10 2016

Prog

(PARI) g(n, r) = for(m=1, n, print1(fib(m, r)", ")) fib(n, r) = x=(sqrt(r)+1)/2; floor((x^n-(1-x)^n)/sqrt(r)+.5)

Crossrefs

Sequence in context: A292200 A249576 A097600 * A173674 A018128 A032189

Adjacent sequences: A136419 A136420 A136421 * A136423 A136424 A136425

Keyword

nonn

Author

Cino Hilliard, Apr 01 2008

Extensions

a(3) corrected by R. J. Mathar, Sep 10 2016

Status

approved