A136421 a(n) = floor((x^n - (1-x)^n)/sqrt(2)+ 1/2) where x = (sqrt(2)+1)/2.

1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 21, 25, 31, 37, 44, 54, 65, 78, 94, 114, 138, 166, 200, 242, 292, 352, 425, 514, 620, 748, 903, 1090, 1316, 1589, 1918, 2315, 2794, 3373, 4072, 4915, 5933, 7162, 8645, 10436, 12597, 15206, 18355, 22156, 26745

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Programs

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

Mathematica

Table[Floor[Fibonacci[n, 2]/2^(n-1) +1/2], {n,1,50}] (* G. C. Greubel, Oct 02 2018 *)

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)

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