A136423 Floor((x^n - (1-x)^n)/2 +.5) where x = (sqrt(4)+1)/2 = 3/2.

1, 1, 2, 3, 4, 6, 9, 13, 19, 29, 43, 65, 97, 146, 219, 328, 493, 739, 1108, 1663, 2494, 3741, 5611, 8417, 12626, 18938, 28408, 42611, 63917, 95876, 143813, 215720, 323580, 485370, 728055, 1092082, 1638123, 2457185, 3685777, 5528666, 8292999

Offset: 1

Author

Cino Hilliard, Apr 01 2008

Keywords

• nonn

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(4)+1)/2=3/2, a(n)/a(n-1) -> x.

Programs

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)

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^(n-1)+A152011(n). - R. J. Mathar, Sep 10 2016